evaluation of measurement data

First edition 2008

© JCGM 2008

JCGM 101:2008

Evaluation of measurement data — Supplement 1 to the “Guide to the expression of uncertainty in measurement” — Propagation of distributions using a Monte Carlo method

Évaluation des données de mesure — Supplément 1 du “Guide pour l’expression de l’incertitude de mesure” — Propagation de distributions par une méthode de Monte Carlo

JCGM 101:2008

c© JCGM 2008

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Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Normative references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Terms and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.1 Main stages of uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.3 Obtaining summary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4 Implementations of the propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.5 Reporting the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.6 GUM uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.7 Conditions for valid application of the GUM uncertainty framework for linear models . . . . . . . . . . . . . . 125.8 Conditions for valid application of the GUM uncertainty framework for non-linear models . . . . . . . . . . 135.9 Monte Carlo approach to the propagation and summarizing stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.10 Conditions for the valid application of the described Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . 145.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method . . . . . . . . . . . 16

6 Probability density functions for the input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Bayes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3 Principle of maximum entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4 Probability density function assignment for some common circumstances . . . . . . . . . . . . . . . . . . . . . . . 19

6.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4.2 Rectangular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4.3 Rectangular distributions with inexactly prescribed limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4.4 Trapezoidal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4.5 Triangular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4.6 Arc sine (U-shaped) distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4.7 Gaussian distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4.8 Multivariate Gaussian distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4.9 t-distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4.10 Exponential distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4.11 Gamma distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.5 Probability distributions from previous uncertainty calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Implementation of a Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Number of Monte Carlo trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.3 Sampling from probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 Evaluation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.5 Discrete representation of the distribution function for the output quantity . . . . . . . . . . . . . . . . . . . . . 297.6 Estimate of the output quantity and the associated standard uncertainty . . . . . . . . . . . . . . . . . . . . . . . 297.7 Coverage interval for the output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.8 Computation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.9 Adaptive Monte Carlo procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.9.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.9.2 Numerical tolerance associated with a numerical value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.9.3 Objective of adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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7.9.4 Adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Validation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.1 Validation of the GUM uncertainty framework using a Monte Carlo method . . . . . . . . . . . . . . . . . . . . 338.2 Obtaining results from a Monte Carlo method for validation purposes . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.1 Illustrations of aspects of this Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.2 Additive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2.2 Normally distributed input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2.3 Rectangularly distributed input quantities with the same width . . . . . . . . . . . . . . . . . . . . . . . . 379.2.4 Rectangularly distributed input quantities with different widths . . . . . . . . . . . . . . . . . . . . . . . . 38

9.3 Mass calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.2 Propagation and summarizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9.4 Comparison loss in microwave power meter calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.2 Propagation and summarizing: zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4.3 Propagation and summarizing: non-zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

9.5 Gauge block calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.5.1 Formulation: model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.5.2 Formulation: assignment of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.5.3 Propagation and summarizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Annexes

A Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Sensitivity coefficients and uncertainty budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

C Sampling from probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.2 General distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.3 Rectangular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

C.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.3.2 Randomness tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60C.3.3 Procedure for generating pseudo-random numbers from a rectangular distribution . . . . . . . . . . . 61

C.4 Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.5 Multivariate Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.6 t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

D Continuous approximation to the distribution function for the output quantity . . . . . . . . . . . . . . 64

E Coverage interval for the four-fold convolution of a rectangular distribution . . . . . . . . . . . . . . . . 66

F Comparison loss problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68F.1 Expectation and standard deviation obtained analytically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68F.2 Analytic solution for zero estimate of the voltage reflection coefficient having associated zero covariance 69F.3 GUM uncertainty framework applied to the comparison loss problem . . . . . . . . . . . . . . . . . . . . . . . . . . 70

F.3.1 Uncorrelated input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70F.3.2 Correlated input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

G Glossary of principal symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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Alphabetical index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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Foreword

In 1997 a Joint Committee for Guides in Metrology (JCGM), chaired by the Director of the Bureau International desPoids et Mesures (BIPM), was created by the seven international organizations that had originally in 1993 preparedthe “Guide to the expression of uncertainty in measurement” (GUM) and the “International vocabulary of basicand general terms in metrology” (VIM). The JCGM assumed responsibility for these two documents from the ISOTechnical Advisory Group 4 (TAG4).

The Joint Committee is formed by the BIPM with the International Electrotechnical Commission (IEC), the Interna-tional Federation of Clinical Chemistry and Laboratory Medicine (IFCC), the International Laboratory AccreditationCooperation (ILAC), the International Organization for Standardization (ISO), the International Union of Pure andApplied Chemistry (IUPAC), the International Union of Pure and Applied Physics (IUPAP), and the InternationalOrganization of Legal Metrology (OIML).

JCGM has two Working Groups. Working Group 1, “Expression of uncertainty in measurement”, has the task topromote the use of the GUM and to prepare Supplements and other documents for its broad application. WorkingGroup 2, “Working Group on International vocabulary of basic and general terms in metrology (VIM)”, has the taskto revise and promote the use of the VIM.

Supplements such as this one are intended to give added value to the GUM by providing guidance on aspects ofuncertainty evaluation that are not explicitly treated in the GUM. The guidance will, however, be as consistent aspossible with the general probabilistic basis of the GUM.

The present Supplement 1 to the GUM has been prepared by Working Group 1 of the JCGM, and has benefited fromdetailed reviews undertaken by member organizations of the JCGM and National Metrology Institutes.

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http://www.bipm.org/en/committees/jc/jcgm/

http://www.bipm.org/

http://www.iec.ch/

http://www.ifcc.org/

http://www.ilac.org/

http://www.iso.org/

http://www.iupac.org/

http://www.iupap.org/

http://www.oiml.org/

JCGM 101:2008

Introduction

This Supplement to the “Guide to the expression of uncertainty in measurement” (GUM) is concerned with thepropagation of probability distributions through a mathematical model of measurement [GUM:1995 3.1.6] as a basisfor the evaluation of uncertainty of measurement, and its implementation by a Monte Carlo method. The treatmentapplies to a model having any number of input quantities, and a single output quantity.

The described Monte Carlo method is a practical alternative to the GUM uncertainty framework [GUM:1995 3.4.8].It has value when

a) linearization of the model provides an inadequate representation, or

b) the probability density function (PDF) for the output quantity departs appreciably from a Gaussian distributionor a scaled and shifted t-distribution, e.g. due to marked asymmetry.

In case a), the estimate of the output quantity and the associated standard uncertainty provided by the GUM un-certainty framework might be unreliable. In case b), unrealistic coverage intervals (a generalization of “expandeduncertainty” in the GUM uncertainty framework) might be the outcome.

The GUM [GUM:1995 3.4.8] “. . . provides a framework for assessing uncertainty . . . ”, based on the law of propagationof uncertainty [GUM:1995 5] and the characterization of the output quantity by a Gaussian distribution or a scaledand shifted t-distribution [GUM:1995 G.6.2, G.6.4]. Within that framework, the law of propagation of uncertaintyprovides a means for propagating uncertainties through the model. Specifically, it evaluates the standard uncertaintyassociated with an estimate of the output quantity, given

1) best estimates of the input quantities,

2) the standard uncertainties associated with these estimates, and, where appropriate,

3) degrees of freedom associated with these standard uncertainties, and

4) any non-zero covariances associated with pairs of these estimates.

Also within the framework, the PDF taken to characterize the output quantity is used to provide a coverage interval,for a stipulated coverage probability, for that quantity.

The best estimates, standard uncertainties, covariances and degrees of freedom summarize the information availableconcerning the input quantities. With the approach considered here, the available information is encoded in termsof PDFs for the input quantities. The approach operates with these PDFs in order to determine the PDF for theoutput quantity.

Whereas there are some limitations to the GUM uncertainty framework, the propagation of distributions will alwaysprovide a PDF for the output quantity that is consistent with the model of the measurement and the PDFs for theinput quantities. This PDF for the output quantity describes the knowledge of that quantity, based on the knowledgeof the input quantities, as described by the PDFs assigned to them. Once the PDF for the output quantity is available,that quantity can be summarized by its expectation, taken as an estimate of the quantity, and its standard deviation,taken as the standard uncertainty associated with the estimate. Further, the PDF can be used to obtain a coverageinterval, corresponding to a stipulated coverage probability, for the output quantity.

The use of PDFs as described in this Supplement is generally consistent with the concepts underlying the GUM.The PDF for a quantity expresses the state of knowledge about the quantity, i.e. it quantifies the degree of beliefabout the values that can be assigned to the quantity based on the available information. The information usuallyconsists of raw statistical data, results of measurement, or other relevant scientific statements, as well as professionaljudgement.

In order to construct a PDF for a quantity, on the basis of a series of indications, Bayes’ theorem can be applied [27, 33].When appropriate information is available concerning systematic effects, the principle of maximum entropy can be

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used to assign a suitable PDF [51, 56].

The propagation of distributions has wider application than the GUM uncertainty framework. It works with richerinformation than that conveyed by best estimates and the associated standard uncertainties (and degrees of freedomand covariances when appropriate).

An historical perspective is given in annex A.

NOTE 1 Citations of the form [GUM:1995 3.1.6] are to the indicated (sub)clauses of the GUM.

NOTE 2 The GUM provides an approach when linearization is inadequate [GUM:1995 5.1.2 note]. The approach has limi-tations: only the leading non-linear terms in the Taylor series expansion of the model are used, and the PDFs for the inputquantities are regarded as Gaussian.

NOTE 3 Strictly, the GUM characterizes the variable (Y − y)/u(y) by a t-distribution, where Y is the output quantity, y anestimate of Y , and u(y) the standard uncertainty associated with y [GUM:1995 G.3.1]. This characterization is also used inthis Supplement. (The GUM in fact refers to the variable (y − Y )/u(y).)

NOTE 4 A PDF for a quantity is not to be understood as a frequency density.

NOTE 5 “The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailedknowledge of the nature of the measurand and of the measurement method and procedure used. The quality and utility of theuncertainty quoted for the result of a measurement therefore ultimately depends on the understanding, critical analysis, andintegrity of those who contribute to the assignment of its value.” [17].

viii c© JCGM 2008— All rights reserved

JCGM 101:2008

Evaluation of measurement data — Supplement 1 to the“Guide to the expression of uncertainty in measurement” —Propagation of distributions using a Monte Carlo method

1 Scope

This Supplement provides a general numerical approach, consistent with the broad principles ofthe GUM [GUM:1995 G.1.5], for carrying out the calculations required as part of an evaluation of measure-ment uncertainty. The approach applies to arbitrary models having a single output quantity where the inputquantities are characterized by any specified PDFs [GUM:1995 G.1.4, G.5.3].

As in the GUM, this Supplement is primarily concerned with the expression of uncertainty in the measurement of a well-defined physical quantity—the measurand—that can be characterized by an essentially unique value [GUM:1995 1.2].

This Supplement also provides guidance in situations where the conditions for the GUM uncertainty frame-work [GUM:1995 G.6.6] are not fulfilled, or it is unclear whether they are fulfilled. It can be used when it is difficultto apply the GUM uncertainty framework, because of the complexity of the model, for example. Guidance is given ina form suitable for computer implementation.

This Supplement can be used to provide (a representation of) the PDF for the output quantity from which

a) an estimate of the output quantity,

b) the standard uncertainty associated with this estimate, and

c) a coverage interval for that quantity, corresponding to a specified coverage probability

can be obtained.

Given (i) the model relating the input quantities and the output quantity and (ii) the PDFs characterizing the inputquantities, there is a unique PDF for the output quantity. Generally, the latter PDF cannot be determined analytically.Therefore, the objective of the approach described here is to determine a), b), and c) above to a prescribed numericaltolerance, without making unquantified approximations.

For a prescribed coverage probability, this Supplement can be used to provide any required coverage interval, includingthe probabilistically symmetric coverage interval and the shortest coverage interval.

This Supplement applies to input quantities that are independent, where each such quantity is assigned an appropri-ate PDF, or not independent, i.e. when some or all of these quantities are assigned a joint PDF.

Typical of the uncertainty evaluation problems to which this Supplement can be applied include those in which

— the contributory uncertainties are not of approximately the same magnitude [GUM:1995 G.2.2],

— it is difficult or inconvenient to provide the partial derivatives of the model, as needed by the law of propagationof uncertainty [GUM:1995 5],

— the PDF for the output quantity is not a Gaussian distribution or a scaled and shifted t-distribution [GUM:1995 G.6.5],

— an estimate of the output quantity and the associated standard uncertainty are approximately of the samemagnitude [GUM:1995 G.2.1],

— the models are arbitrarily complicated [GUM:1995 G.1.5], and

c© JCGM 2008— All rights reserved 1

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— the PDFs for the input quantities are asymmetric [GUM:1995 G.5.3].

A validation procedure is provided to check whether the GUM uncertainty framework is applicable. The GUM uncer-tainty framework remains the primary approach to uncertainty evaluation in circumstances where it is demonstrablyapplicable.

It is usually sufficient to report measurement uncertainty to one or perhaps two significant decimal digits. Guidanceis provided on carrying out the calculation to give reasonable assurance that in terms of the information provided thereported decimal digits are correct.

Detailed examples illustrate the guidance provided.

This document is a Supplement to the GUM and is to be used in conjunction with it. Other approaches generallyconsistent with the GUM may alternatively be used. The audience of this Supplement is that of the GUM.

NOTE 1 This Supplement does not consider models that do not define the output quantity uniquely (for example, involvingthe solution of a quadratic equation, without specifying which root is to be taken).

NOTE 2 This Supplement does not consider the case where a prior PDF for the output quantity is available, but the treatmenthere can be adapted to cover this case [16].

2 Normative references

The following referenced documents are indispensable for the application of this document.

JCGM 100 (GUM:1995). Guide to the expression of uncertainty in measurement (GUM), 1995.

JCGM 200 (VIM:2008). International Vocabulary of Metrology—Basic and General Concepts and Associated Terms,VIM, 3rd Edition, 2008.

3 Terms and definitions

For the purposes of this document the terms and definitions of the GUM and the “International vocabulary of basicand general terms in metrology” (VIM) apply unless otherwise indicated. Some of the most relevant definitions,adapted where necessary from these documents (see 4.2), are given below. Further definitions are given, includingdefinitions taken or adapted from other sources, that are important for this Supplement.

A glossary of principal symbols is given in annex G.

3.1probability distribution〈random variable〉 function giving the probability that a random variable takes any given value or belongs to a givenset of values

NOTE The probability on the whole set of values of the random variable equals 1.

[Adapted from ISO 3534-1:1993 1.3; GUM:1995 C.2.3]

NOTE 1 A probability distribution is termed univariate when it relates to a single (scalar) random variable, and multivariatewhen it relates to a vector of random variables. A multivariate probability distribution is also described as a joint distribution.

NOTE 2 A probability distribution can take the form of a distribution function or a probability density function.

2 c© JCGM 2008— All rights reserved

http://www.bipm.org/en/publications/guides/vim.html

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3.2distribution functionfunction giving, for every value ξ, the probability that the random variable X be less than or equal to ξ:

GX(ξ) = Pr(X ≤ ξ)


3.3probability density functionderivative, when it exists, of the distribution function

gX(ξ) = dGX(ξ)/dξ

NOTE gX(ξ) dξ is the “probability element”

gX(ξ) dξ = Pr(ξ < X < ξ + dξ).


3.4normal distributionprobability distribution of a continuous random variable X having the probability density function

gX(ξ) =1

σ√

2πexp

[−1

2

(ξ − µ

σ

)2]

,

for −∞ < ξ < +∞

NOTE µ is the expectation and σ is the standard deviation of X.


NOTE The normal distribution is also known as a Gaussian distribution.

3.5t-distributionprobability distribution of a continuous random variable X having the probability density function

gX(ξ) =Γ((ν + 1)/2)√

πνΓ(ν/2)

(1 +

ξ2

ν

)−(ν+1)/2

,

for −∞ < ξ < +∞, with parameter ν, a positive integer, the degrees of freedom of the distribution, where

Γ(z) =∫ ∞

0

tz−1e−t dt, z > 0,

is the gamma function

3.6expectationproperty of a random variable, which, for a continuous random variable X characterized by a PDF gX(ξ), is given by

E(X) =∫ ∞

−∞ξgX(ξ) dξ

NOTE 1 Not all random variables have an expectation.


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NOTE 2 The expectation of the random variable Z = F (X), for a given function F (X), is

E(Z) = E(F (X)) =

∫ ∞

−∞F (ξ)gX(ξ) dξ.

3.7varianceproperty of a random variable, which, for a continuous random variable X characterized by a PDF gX(ξ), is given by

V (X) =∫ ∞

−∞[ξ − E(X)]2gX(ξ) dξ

NOTE Not all random variables have a variance.

3.8standard deviationpositive square root [V (X)]1/2 of the variance

3.9moment of order rexpectation of the rth power of a random variable, namely

E(Xr) =∫ ∞

−∞ξrgX(ξ) dξ

NOTE 1 The central moment of order r is the expectation of the random variable Z = [X − E(X)]r.

NOTE 2 The expectation E(X) is the first moment. The variance V (X) is the central moment of order 2.

3.10covarianceproperty of a pair of random variables, which, for two continuous random variables X1 and X2 characterized by ajoint (multivariate) PDF gX(ξ), where X = (X1, X2)> and ξ = (ξ1, ξ2)>, is given by

Cov(X1, X2) =∫ ∞

−∞

∫ ∞

−∞[ξ1 − E(X1)][ξ2 − E(X2)]gX(ξ) dξ1 dξ2

NOTE Not all pairs of random variables have a covariance.

3.11uncertainty matrixmatrix of dimension N × N , containing on its diagonal the squares of the standard uncertainties associated withestimates of the components of an N -dimensional vector quantity, and in its off-diagonal positions the covariancesassociated with pairs of estimates

NOTE 1 An uncertainty matrix Ux of dimension N × N associated with the vector estimate x of a vector quantity X hasthe representation

Ux =

u(x1, x1) · · · u(x1, xN )...

. . ....

u(xN , x1) · · · u(xN , xN )

,

where u(xi, xi) = u2(xi) is the variance (squared standard uncertainty) associated with xi and u(xi, xj) is the covarianceassociated with xi and xj . u(xi, xj) = 0 if elements Xi and Xj of X are uncorrelated.

NOTE 2 Covariances are also known as mutual uncertainties.

NOTE 3 An uncertainty matrix is also known as a covariance matrix or variance-covariance matrix.


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3.12coverage intervalinterval containing the value of a quantity with a stated probability, based on the information available

NOTE 1 A coverage interval is sometimes known as a credible interval or a Bayesian interval.

NOTE 2 Generally there is more than one coverage interval for a stated probability.

NOTE 3 A coverage interval should not be termed ‘confidence interval’ to avoid confusion with the statistical con-cept [GUM:1995 6.2.2].

NOTE 4 This definition differs from that in the VIM, 3rd Edition (2008), since the term ‘true value’ has not been used in thisSupplement, for reasons given in the GUM [GUM:1995 E.5].

3.13coverage probabilityprobability that the value of a quantity is contained within a specified coverage interval

NOTE The coverage probability is sometimes termed “level of confidence” [GUM:1995 6.2.2].

3.14length of a coverage intervallargest value minus smallest value in a coverage interval

3.15probabilistically symmetric coverage intervalcoverage interval for a quantity such that the probability that the quantity is less than the smallest value in the intervalis equal to the probability that the quantity is greater than the largest value in the interval

3.16shortest coverage intervalcoverage interval for a quantity with the shortest length among all coverage intervals for that quantity having thesame coverage probability

3.17propagation of distributionsmethod used to determine the probability distribution for an output quantity from the probability distributionsassigned to the input quantities on which the output quantity depends

NOTE The method may be analytical or numerical, exact or approximate.

3.18GUM uncertainty frameworkapplication of the law of propagation of uncertainty and the characterization of the output quantity by a Gaussiandistribution or a scaled and shifted t-distribution in order to provide a coverage interval

3.19Monte Carlo methodmethod for the propagation of distributions by performing random sampling from probability distributions

3.20numerical tolerancesemi-width of the shortest interval containing all numbers that can correctly be expressed to a specified number ofsignificant decimal digits

EXAMPLE All numbers greater than 1.75 and less than 1.85 can be expressed to two significant decimal digits as 1.8. Thenumerical tolerance is (1.85− 1.75)/2 = 0.05.

NOTE For the calculation of numerical tolerance associated with a numerical value, see 7.9.2.


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4 Conventions and notation

For the purposes of this Supplement the following conventions and notation are adopted.

4.1 A mathematical model of a measurement [GUM:1995 4.1] of a single (scalar) quantity can be expressed as afunctional relationship f :

Y = f(X), (1)

where Y is a scalar output quantity and X represents the N input quantities (X1, . . . , XN )>. Each Xi is regardedas a random variable with possible values ξi and expectation xi. Y is a random variable with possible values η andexpectation y.

NOTE 1 The same symbol is used for a physical quantity and the random variable that represents that quantity(cf. [GUM:1995 4.1.1 note 1]).

NOTE 2 Many models of measurement can be expressed in the form (1). A more general form is

h(Y, X) = 0,

which implicitly relates X and Y . In any case, to apply the described Monte Carlo method, it is only necessary that Y can beformed corresponding to any meaningful X.

4.2 This Supplement departs from the symbols often used for ‘PDF’ and ‘distribution function’ [24]. The GUMuses the generic symbol f to refer to a model and a PDF. Little confusion arises in the GUM as a consequence ofthis usage. The situation in this Supplement is different. The concepts of model, PDF, and distribution functionare central to following and implementing the guidance provided. Therefore, in place of the symbols f and F todenote a PDF and a distribution function, respectively, the symbols g and G are used. These symbols are indexedappropriately to denote the quantity concerned. The symbol f is reserved for the model.

NOTE The definitions in clause 3 that relate to PDFs and distributions are adapted accordingly.

4.3 In this Supplement, a PDF is assigned to a quantity, which may be a single, scalar quantity X or a vectorquantity X. In the scalar case, the PDF for X is denoted by gX(ξ), where ξ is a variable describing the possible valuesof X. This X is considered as a random variable with expectation E(X) and variance V (X) (see 3.6 and 3.7).

4.4 In the vector case, the PDF for X is denoted by gX(ξ), where ξ = (ξ1, . . . , ξN )> is a vector variable describ-ing the possible values of the vector quantity X. This X is considered as a random vector variable with (vector)expectation E(X) and covariance matrix V (X).

4.5 A PDF for more than one input quantity is often called joint even if all the input quantities are independent.

4.6 When the elements Xi of X are independent, the PDF for Xi is denoted by gXi(ξi).

4.7 The PDF for Y is denoted by gY (η) and the distribution function for Y by GY (η).

4.8 In the body of this Supplement, a quantity is generally denoted by an upper case letter and the expectation ofthe quantity or an estimate of the quantity by the corresponding lower case letter. For example, the expectation oran estimate of a quantity Y would be denoted by y. Such a notation is largely inappropriate for physical quantities,because of the established use of specific symbols, e.g. T for temperature and t for time. Therefore, in some of theexamples (clause 9), a different notation is used. There, a quantity is denoted by its conventional symbol and itsexpectation or an estimate of it by that symbol hatted. For instance, the quantity representing the deviation of thelength of a gauge block being calibrated from nominal length (see 9.5) is denoted by δL and an estimate of δL by δL.

NOTE A hatted symbol is generally used in the statistical literature to denote an estimate.

4.9 In this Supplement, the term “law of propagation of uncertainty” applies to the use of a first-order Taylor seriesapproximation to the model. The term is qualified accordingly when a higher-order approximation is used.


JCGM 101:2008

4.10 The subscript “c” [GUM:1995 5.1.1] for the combined standard uncertainty is redundant in this Supplement.The standard uncertainty associated with an estimate y of an output quantity Y can therefore be written as u(y),but the use of uc(y) remains acceptable if it is helpful to emphasize the fact that it represents a combined standarduncertainty. The qualifier “combined” in this context is also regarded as superfluous and may be omitted: the presenceof “y” in “u(y)” already indicates the estimate with which the standard uncertainty is associated. Moreover, whenthe results of one or more uncertainty evaluations become inputs to a subsequent uncertainty evaluation, the use ofthe subscript “c” and the qualifier “combined” are then inappropriate.

4.11 The terms “coverage interval” and “coverage probability” are used throughout this Supplement. The GUMuses the term “level of confidence” as a synonym for coverage probability, drawing a distinction between “level ofconfidence” and “confidence level” [GUM:1995 6.2.2], because the latter has a specific definition in statistics. Since,in some languages, the translation from English of these two terms yields the same expression, the use of these termsis avoided here.

4.12 According to Resolution 10 of the 22nd CGPM (2003) “ . . . the symbol for the decimal marker shall be eitherthe point on the line or the comma on the line . . . ”. The JCGM has decided to adopt, in its documents in English,the point on the line.

4.13 Unless otherwise qualified, numbers are expressed in a manner that indicates the number of meaningfulsignificant decimal digits.

EXAMPLE The numbers 0.060, 0.60, 6.0 and 60 are expressed to two significant decimal digits. The numbers 0.06, 0.6, 6and 6× 101 are expressed to one significant decimal digit. It would be incorrect to express 6× 101 as 60, since two significantdecimal digits would be implied.

4.14 Some symbols have more than one meaning in this Supplement. See annex G. The context clarifies the usage.

4.15 The following abbreviations are used in this Supplement:

CGPM Conference Generale des Poids et MesuresIEEE Institute of Electrical and Electronic EngineersGUF GUM uncertainty frameworkJCGM Joint Committee for Guides in MetrologyGUM Guide to the expression of uncertainty in measurementMCM Monte Carlo methodPDF probability density functionVIM International vocabulary of basic and general terms in metrology

5 Basic principles

5.1 Main stages of uncertainty evaluation

5.1.1 The main stages of uncertainty evaluation constitute formulation, propagation, and summarizing:

a) Formulation:

1) define the output quantity Y , the quantity intended to be measured (the measurand);

2) determine the input quantities X = (X1, . . . , XN )> upon which Y depends;

3) develop a model relating Y and X;

4) on the basis of available knowledge assign PDFs—Gaussian (normal), rectangular (uniform), etc.—to the Xi.Assign instead a joint PDF to those Xi that are not independent;

b) Propagation: propagate the PDFs for the Xi through the model to obtain the PDF for Y ;


JCGM 101:2008

c) Summarizing: use the PDF for Y to obtain

1) the expectation of Y , taken as an estimate y of the quantity,

2) the standard deviation of Y , taken as the standard uncertainty u(y) associated with y [GUM:1995 E.3.2],and

3) a coverage interval containing Y with a specified probability (the coverage probability).

NOTE 1 The expectation may not be appropriate for all applications (cf. [GUM:1995 4.1.4]).

NOTE 2 The quantities described by some distributions, such as the Cauchy distribution, have no expectation or standarddeviation. A coverage interval for the output quantity can always be obtained, however.

5.1.2 The GUM uncertainty framework does not explicitly refer to the assignment of PDFs to the input quanti-ties. However [GUM:1995 3.3.5], “. . . a Type A standard uncertainty is obtained from a probability density function. . . derived from an observed frequency distribution . . . , while a Type B standard uncertainty is obtained from anassumed probability density function based on the degree of belief that an event will occur . . . . Both approachesemploy recognized interpretations of probability.”

NOTE The use of probability distributions in a Type B evaluation of uncertainty is a feature of Bayesian inference [21, 27].Research continues [22] on the boundaries of validity for the assignment of degrees of freedom to a standard uncertainty basedon the Welch-Satterthwaite formula.

5.1.3 The steps in the formulation stage are carried out by the metrologist, perhaps with expert support. Guidanceon the assignment of PDFs (step 4) of stage a) in 5.1.1) is given in this Supplement for some common cases (see 6.4).The propagation and summarizing stages, b) and c) in 5.1.1, for which detailed guidance is provided here, requireno further metrological information, and in principle can be carried out to any required numerical tolerance for theproblem specified in the formulation stage.

NOTE Once the formulation stage a) in 5.1.1 has been carried out, the PDF for the output quantity is completely specifiedmathematically, but generally the calculation of the expectation, standard deviation and coverage intervals requires numericalmethods that involve a degree of approximation.

5.2 Propagation of distributions

In this Supplement a generally efficient approach for determining (a numerical approximation to) the distributionfunction

GY (η) =∫ η

−∞gY (z) dz

for Y is considered. It is based on applying a Monte Carlo method (MCM) as an implementation of the propagationof distributions (see 5.9).

NOTE A formal definition [9] for the PDF for Y is

gY (η) =

∫ ∞

−∞· · ·∫ ∞

−∞gX(ξ)δ(η − f(ξ)) dξN · · ·dξ1,

where δ(·) denotes the Dirac delta function. This multiple integral cannot generally be evaluated analytically. A numericalintegration rule can be applied to provide an approximation to gY (η), but this is not an efficient approach.

5.3 Obtaining summary information

5.3.1 An estimate y of Y is the expectation E(Y ). The standard uncertainty u(y) associated with y is given bythe standard deviation of Y , the positive square root of the variance V (Y ) of Y .


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5.3.2 A coverage interval for Y can be determined from GY (η). Let α denote any numerical value between zeroand 1−p, where p is the required coverage probability. The endpoints of a 100p % coverage interval for Y are G−1

Y (α)and G−1

Y (p + α), i.e. the α- and (p + α)-quantiles of GY (η).

5.3.3 The choice α = (1 − p)/2 gives the coverage interval defined by the (1 − p)/2- and (1 + p)/2-quantiles,providing a probabilistically symmetric 100p % coverage interval.

NOTE When the PDF for Y is symmetric about the estimate y, the coverage interval obtained would be identical to y ± Up,where the expanded uncertainty [GUM:1995 2.3.5] Up is given by the product of the standard uncertainty u(y) and the coveragefactor that is appropriate for that PDF. This PDF is generally not known analytically.

5.3.4 A numerical value of α different from (1 − p)/2 may be more appropriate if the PDF is asymmetric.The shortest 100p % coverage interval can be used in this case. It has the property that, for a unimodal (single-peaked) PDF, it contains the mode, the most probable value of Y . It is given by the numerical value of α sat-isfying gY (G−1

Y (α)) = gY (G−1Y (p + α)), if gY (η) is unimodal, and in general by the numerical value of α such that

G−1Y (p + α)−G−1

Y (α) is a minimum.

5.3.5 The probabilistically symmetric 100p % coverage interval and the shortest 100p % coverage interval areidentical for a symmetric PDF, such as the Gaussian and scaled and shifted t-distribution used within the GUMuncertainty framework. Therefore, in comparing the GUM uncertainty framework with other approaches, either ofthese intervals can be used.

5.3.6 Figure 1 shows the distribution function GY (η) corresponding to an asymmetric PDF. Broken vertical linesmark the endpoints of the probabilistically symmetric 95 % coverage interval and broken horizontal lines the corre-sponding probability points, viz. 0.025 and 0.975. Continuous lines mark the endpoints of the shortest 95 % coverageinterval and the corresponding probability points, which are 0.006 and 0.956 in this case. The lengths of the 95 %coverage intervals in the two cases are 1.76 unit and 1.69 unit, respectively.

Figure 1 — A distribution function GY (η) corresponding to an asymmetric PDF and the probabilistically

symmetric and shortest 95 % coverage intervals (5.3.6). “Unit” denotes any unit

5.4 Implementations of the propagation of distributions

5.4.1 The propagation of distributions can be implemented in several ways:

a) analytical methods, i.e. methods that provide a mathematical representation of the PDF for Y ;

b) uncertainty propagation based on replacing the model by a first-order Taylor series approxima-tion [GUM:1995 5.1.2] — the law of propagation of uncertainty;


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c) as b), except that contributions derived from higher-order terms in the Taylor series approximation are in-cluded [GUM:1995 5.1.2 note];

d) numerical methods [GUM:1995 G.1.5] that implement the propagation of distributions, specifically using MCM(see 5.9).

NOTE 1 Analytical methods are ideal in that they do not introduce any approximation. They are applicable in simple casesonly, however. A treatment and examples are available [8, 13]. These methods are not considered further in this Supplement,apart from in the examples (clause 9) for comparison purposes.

NOTE 2 MCM as considered here is regarded as a means for providing a numerical representation of the distribution for theoutput quantity, rather than a simulation method per se. In the context of the propagation stage of uncertainty evaluation, theproblem to be solved is deterministic, there being no random physical process to be simulated.

5.4.2 Approaches to uncertainty evaluation other than the GUM uncertainty framework are permitted bythe GUM [GUM:1995 G.1.5]. The approach advocated in this Supplement, based on the propagation of distribu-tions, is general. For linear or linearized models and input quantities for which the PDFs are Gaussian, the approachyields results consistent with the GUM uncertainty framework. However, in cases where the conditions for the GUMuncertainty framework to be applied (see 5.7 and 5.8) do not hold, the approach of this Supplement can generally beexpected to lead to a valid uncertainty statement.

5.4.3 An appropriate method has to be chosen for the propagation stage. If it can be demonstrated that theconditions necessary for the GUM uncertainty framework to give valid results hold, then that approach can be used.If there are indications that the GUM uncertainty framework is likely to be invalid, then another approach should beemployed. A third situation can arise in which it is difficult to assess whether or not the GUM uncertainty frameworkwill be valid. In all three cases, MCM provides a practical (alternative) method. In the first case, MCM maysometimes be easier to apply due to difficulties in calculating sensitivity coefficients [GUM:1995 5.1.3], for example. Inthe second, MCM can generally be expected to give valid results, since it does not make approximating assumptions.In the third, MCM can be applied either to determine the results directly or to assess the quality of those providedby the GUM uncertainty framework.

5.4.4 The propagation of the PDFs gXi(ξi), i = 1, . . . , N , for the input quantities Xi through the model to provide

the PDF gY (η) for the output quantity Y is illustrated in figure 2 for N = 3 independent Xi. Figure 2 may becompared to figure 3 for the law of propagation of uncertainty. In figure 2, the gXi

(ξi), i = 1, 2, 3, are Gaussian,triangular, and Gaussian, respectively. gY (η) is indicated as being asymmetric, as generally arises for non-linearmodels or asymmetric gXi

(ξi).

-

gX3(ξ3)

-

gX2(ξ2)

-

gX1(ξ1)

Y = f(X) -

gY (η)

Figure 2 — Illustration of the propagation of distributions for N = 3 independent input quantities (5.4.4)

5.4.5 In practice, only for simple cases can the propagation of distributions be implemented without makingapproximations. The GUM uncertainty framework implements one approximate method, and MCM another. For asmall but important subset of problems, the GUM uncertainty framework is exact. MCM is never exact, but is morevalid than the GUM uncertainty framework for a large class of problems.

5.5 Reporting the results

5.5.1 The following items would typically be reported following the use of the propagation of distributions:


JCGM 101:2008

a) an estimate y of the output quantity Y ;

b) the standard uncertainty u(y) associated with y;

c) the stipulated coverage probability 100p % (e.g. 95 %);

d) the endpoints of the selected 100p % coverage interval (e.g. 95 % coverage interval) for Y ;

e) any other relevant information, such as whether the coverage interval is a probabilistically symmetric coverageinterval or a shortest coverage interval.

5.5.2 y, u(y) and the endpoints of a 100p % coverage interval for Y should be reported to a number of decimaldigits such that the least significant decimal digit is in the same position with respect to the decimal point as thatfor u(y) [GUM:1995 7.2.6]. One or two significant decimal digits would usually be adequate to represent u(y).

NOTE 1 Each reported numerical value would typically be obtained by rounding a numerical value expressed to a greaternumber of significant decimal digits.

NOTE 2 A factor influencing the choice of one or two significant decimal digits is the leading significant decimal digit of u(y).If this digit is 1 or 2, the deviation of the reported numerical value of u(y) from its numerical value before rounding is largerelative to the latter numerical value. If the leading significant decimal digit is 9, the deviation is relatively smaller.

NOTE 3 If the results are to be used within further calculations, consideration should be given to whether additional decimaldigits should be retained.

EXAMPLE Reported results corresponding to declaring two significant decimal digits in u(y), for a case in which the coverageinterval is asymmetric with respect to y, are

y = 1.024 V, u(y) = 0.028 V,

shortest 95 % coverage interval = [0.983, 1.088] V.

The same results reported to one significant decimal digit in u(y) would be

y = 1.02 V, u(y) = 0.03 V,

shortest 95 % coverage interval = [0.98, 1.09] V.

5.6 GUM uncertainty framework

5.6.1 The GUM provides general guidance on many aspects of the stages of uncertainty evaluation presentedin 5.1.1. It also provides the GUM uncertainty framework for the propagation and summarizing stages of uncertaintyevaluation. The GUM uncertainty framework has been adopted by many organizations, is widely used, and has beenimplemented in standards and guides on measurement uncertainty and also in software.

5.6.2 The GUM uncertainty framework comprises the following stages. Each model input quantity Xi is sum-marized by its expectation and standard deviation, as given by the PDF for that quantity [GUM:1995 4.1.6]. Theexpectation is taken as the best estimate xi of Xi and the standard deviation as the standard uncertainty u(xi)associated with xi. This information is propagated, using the law of propagation of uncertainty [GUM:1995 5.1.2],through a first- or higher-order Taylor series approximation to the model to provide

a) an estimate y of the output quantity Y , and

b) the standard uncertainty u(y) associated with y.

The estimate y is given by evaluating the model at the xi. A coverage interval for Y is provided based on takingthe PDF for Y as Gaussian or, if the degrees of freedom associated with u(y) is finite [GUM:1995 G], as a scaled andshifted t-distribution.

NOTE The summaries of the Xi also include, where appropriate, the degrees of freedom associated with the u(xi)[GUM:1995 4.2.6]. They also include, where appropriate, covariances associated with pairs of xi [GUM:1995 5.2.5].


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5.6.3 The propagation and summarizing stages of the GUM uncertainty framework (stages b) and c) in 5.1.1)constitute the following computational steps. Also see figure 3, which illustrates the law of propagation of uncertaintyfor a model having N = 3 independent input quantities X = (X1, X2, X3)>, which are estimated by xi with asso-ciated standard uncertainties u(xi), i = 1, 2, 3. The output quantity Y is estimated by y, with associated standarduncertainty u(y).

a) Obtain from the PDFs for the input quantities X = (X1, . . . , XN )> the expectations x = (x1, . . . , xN )> and thestandard deviations (standard uncertainties) u(x) = [u(x1), . . . , u(xN )]>. Use instead the joint PDF for X ifpairs of the Xi are not independent (in which case they have non-zero covariance).

b) Set the degrees of freedom (infinite or finite) associated with each u(xi).

c) For each pair i, j for which Xi and Xj are not independent, obtain from the joint PDF for Xi and Xj thecovariance (mutual uncertainty) u(xi, xj) associated with xi and xj .

d) Form the partial derivatives of first order of f(X) with respect to X.

e) Calculate y, the model evaluated at X equal to x.

f) Calculate the model sensitivity coefficients [GUM:1995 5.1.3] as the above partial derivatives evaluated at x.

g) Calculate the standard uncertainty u(y) by combining u(x), the u(xi, xj), and the model sensitivity coeffi-cients [GUM:1995 formulæ (10), (13)].

h) Calculate νeff , the effective degrees of freedom associated with u(y), using the Welch-Satterthwaite for-mula [GUM:1995 formula (G.2b)].

i) Calculate the expanded uncertainty Up, and hence a coverage interval (for a stipulated coverage probability p)for Y , regarded as a random variable, by forming the appropriate multiple of u(y) through taking the probabilitydistribution of (Y − y)/u(y) as a standard Gaussian distribution (νeff = ∞) or t-distribution (νeff < ∞).

x1, u(x1) -

x2, u(x2) -

x3, u(x3) -

Y = f(X) - y, u(y)

Figure 3 — Illustration of the law of propagation of uncertainty for N = 3 independent input quantities

(5.4.4, 5.6.3)

5.7 Conditions for valid application of the GUM uncertainty framework for linear models

5.7.1 No condition is necessary for the valid application of the law of propagation of uncertainty to linear models(models that are linear in the Xi).

5.7.2 A coverage interval can be determined, in terms of the information provided in the GUM, under the followingconditions:

a) the Welch-Satterthwaite formula is adequate for calculating the effective degrees of freedom associatedwith u(y) [GUM:1995 G.4.1], when one or more of the u(xi) has an associated degrees of freedom that is fi-nite;

b) the Xi are independent when the degrees of freedom associated with the u(xi) are finite;

c) the PDF for Y can adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution.


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NOTE 1 Condition a) is required in order that Y can be characterized by an appropriate scaled and shifted t-distribution.

NOTE 2 Condition b) is required because the GUM does not treat Xi that are not independent in conjunction with finitedegrees of freedom.

NOTE 3 Condition c) is satisfied when each Xi is assigned a Gaussian distribution. It is also satisfied when the conditionsfor the central limit theorem [GUM:1995 G.2] hold.

NOTE 4 The GUM uncertainty framework may not validly be applicable when there is an Xi whose assigned distribution isnon-Gaussian and the corresponding contribution to u(y) is dominant.

5.7.3 When the conditions in 5.7.2 hold, the results from the application of the GUM uncertainty framework canbe expected to be valid for linear models. These conditions apply in many circumstances.

5.8 Conditions for valid application of the GUM uncertainty framework for non-linear models

5.8.1 The law of propagation of uncertainty can validly be applied for non-linear models under the followingconditions:

a) f is continuously differentiable with respect to the elements Xi of X in the neighbourhood of the best estimates xi

of the Xi;

b) condition a) applies for all derivatives up to the appropriate order;

c) the Xi involved in significant higher-order terms of a Taylor series approximation to f(X) are independent;

d) the PDFs assigned to Xi involved in higher-order terms of a Taylor series approximation to f(X) are Gaussian;

e) higher-order terms that are not included in the Taylor series approximation to f(X) are negligible.

NOTE 1 Condition a) is necessary for the applicability of the law of propagation of uncertainty based on a first-order Taylorseries approximation to f(X) when the non-linearity of f is insignificant [GUM:1995 5.1.2].

NOTE 2 Condition b) is necessary for the application of the law of propagation of uncertainty based on a higher-order Taylorseries approximation to f(X) [GUM:1995 5.1.2]. An expression for the most important terms of next highest order to beincluded are given in the GUM [GUM:1995 5.1.2 note].

NOTE 3 Condition c) relates to the statement in the GUM [GUM:1995 5.1.2 note] concerning significant model non-linearityin the case of independent Xi. The GUM does not consider Xi that are not independent in this context.

NOTE 4 Condition d) constitutes a correction to the statement in the GUM [GUM:1995 5.1.2 note] that the version of thelaw of propagation of uncertainty using higher-order terms is based on the symmetry of the PDFs for the Xi [19, 27].

NOTE 5 If the analytical determination of the higher derivatives, required when the non-linearity of the model is significant,is difficult or error-prone, suitable software for automatic differentiation can be used. Alternatively, these derivatives can beapproximated numerically using finite differences [5]. (The GUM provides a finite-difference formula for partial derivatives offirst order [GUM:1995 5.1.3 note 2].) Care should be taken, however, because of the effects of subtractive cancellation whenforming differences between numerically close model values.

5.8.2 A coverage interval can be determined, in terms of the information provided in the GUM, when conditions a),b) and c) in 5.7.2 apply, with the exception that the content of note 3 in that subclause is replaced by “Condition c)is required in order that coverage intervals can be determined from these distributions.”

5.8.3 When the conditions in 5.8.1 and 5.8.2 hold, the results from the application of the GUM uncertaintyframework can be expected to be valid for non-linear models. These conditions apply in many circumstances.

5.9 Monte Carlo approach to the propagation and summarizing stages

5.9.1 MCM provides a general approach to obtain an approximate numerical representation G, say, of the distri-bution function GY (η) for Y [32, page 75]. The heart of the approach is repeated sampling from the PDFs for the Xi


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and the evaluation of the model in each case.

5.9.2 Since GY (η) encodes all the information known about Y , any property of Y such as expectation, varianceand coverage intervals can be approximated using G. The quality of these calculated results improves as the numberof times the PDFs are sampled increases.

5.9.3 Expectations and variances (and higher moments) can be determined directly from the set of model valuesobtained. The determination of coverage intervals requires these model values to be ordered.

5.9.4 If yr, for r = 1, . . . ,M , represent M model values sampled independently from a probability distributionfor Y , then the expectation E(Y ) and variance V (Y ) can be approximated using the yr. In general, the momentsof Y (including E(Y ) and V (Y )) are approximated by those of the sampled model values. Let My0 denote the numberof yr that are no greater than y0, any prescribed number. The probability Pr(Y ≤ y0) is approximated by My0/M .In this way, the yr provide a step function (histogram-like) approximation to the distribution function GY (η).

5.9.5 Each yr is obtained by sampling at random from each of the PDFs for the Xi and evaluating the model atthe sampled values so obtained. G, the primary output from MCM, constitutes the yr arranged in strictly increasingorder.

NOTE It is remotely possible that equalities exist amongst the yr, in which case suitable minute perturbations made to the yr

would enable the yr to be arranged in strictly increasing order. See 7.5.1.

5.9.6 MCM as an implementation of the propagation of distributions is shown diagrammatically in figure 4 for Mprovided in advance (see 7.9 otherwise). MCM can be stated as a step-by-step procedure:

a) select the number M of Monte Carlo trials to be made. See 7.2;

b) generate M vectors, by sampling from the assigned PDFs, as realizations of the (set of N) input quantities Xi.See 7.3;

c) for each such vector, form the corresponding model value of Y , yielding M model values. See 7.4;

d) sort these M model values into strictly increasing order, using the sorted model values to provide G. See 7.5;

e) use G to form an estimate y of Y and the standard uncertainty u(y) associated with y. See 7.6;

f) use G to form an appropriate coverage interval for Y , for a stipulated coverage probability p. See 7.7.

NOTE 1 Subclause 6.4 and annex C provide information on sampling from probability distributions.

NOTE 2 Mathematically, the average of the M model values is a realization of a random variable with expectation E(Y )and variance V (Y )/M . Thus, the closeness of agreement between this average and E(Y ) can be expected to be proportional

to M−1/2.

NOTE 3 Step e) can equally be carried out by using the M model values of Y unsorted. It is necessary to sort these modelvalues to determine the coverage interval in step f).

5.9.7 The effectiveness of MCM to determine y, u(y) and a coverage interval for Y depends on the use of anadequately large value of M (step a) in 5.9.6). Guidance on obtaining such a value and generally on implementing MCMis available [7]. Also see 7.2 and 7.9.

5.10 Conditions for the valid application of the described Monte Carlo method

5.10.1 The propagation of distributions implemented using MCM can validly be applied, and the required summaryinformation subsequently determined, using the approach provided in this Supplement, under the following conditions:

a) f is continuous with respect to the elements Xi of X in the neighbourhood of the best estimates xi of the Xi;


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coverage interval

[ylow, yhigh] for Y

Subclause 7.7

estimate y of Y andassociated standard

uncertainty u(y)

Subclause 7.6

? ?

discrete representation Gof distribution functionfor output quantity Y

Subclause 7.5

?

M model valuesyr = f(xr), r = 1, . . . , M

Subclause 7.4

?

M vectors x1, . . . , xM

sampled from gX(ξ)

Subclause 7.3

?

model Y = f(X)

Subclause 4.1

?

joint PDF gX(ξ) for

input quantities X

Subclause 6

?

number M of MonteCarlo trials

Subclause 7.2

?

coverage probability p

Term 3.13

MCM inputs

MCM propagation:draws from the joint PDFfor the input quantities

and model evaluation forthese draws

primary MCM output:distribution function for

the output quantity

MCM summarizing

Figure 4 — The propagation and summarizing stages of uncertainty evaluation using MCM to implement the

propagation of distributions (5.9.6, 7.1)


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b) the distribution function for Y is continuous and strictly increasing;

c) the PDF for Y is

1) continuous over the interval for which this PDF is strictly positive,

2) unimodal (single-peaked), and

3) strictly increasing (or zero) to the left of the mode and strictly decreasing (or zero) to the right of the mode;

d) E(Y ) and V (Y ) exist;

e) a sufficiently large value of M is used.

NOTE 1 Regarding condition a), no condition on the derivatives of f is required.

NOTE 2 Conditions a) and b) are necessary to ensure that the inverse of the distribution function is unique and hence coverageintervals can be determined. Only condition a) is needed if a coverage interval is not required.

NOTE 3 Condition c) is needed only if the shortest coverage interval is to be determined. In that case, the condition isnecessary to ensure that the shortest coverage interval corresponding to a stipulated coverage probability is unique. The modemay occur at an endpoint of the interval over which this PDF is strictly positive, in which case one of the two conditions in 3)is vacuous.

NOTE 4 Condition d) is needed for (stochastic) convergence of MCM as the number M of trials (see 7.2) increases.

NOTE 5 Condition e) is necessary to ensure that the summarizing information is reliable. See 8.2.

5.10.2 When the conditions in 5.10.1 hold, the results from the application of the propagation of distributionsimplemented in terms of MCM can be expected to be valid. These conditions are less restrictive than those (see 5.7and 5.8) for the application of the GUM uncertainty framework.

5.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method

5.11.1 The intention of this subclause is to compare the principles on which the GUM uncertainty frameworkand MCM as an implementation of the propagation of distributions are based. This subclause also provides some mo-tivation for the use of MCM in circumstances where it is questionable whether the application of the GUM uncertaintyframework is valid.

5.11.2 For the purposes of comparing the GUM uncertainty framework and MCM, it is helpful to review theconsiderations in the GUM regarding Type A and Type B evaluations of uncertainty. For Type A evaluation, the GUMprovides guidance on obtaining a best estimate of a quantity and the associated standard uncertainty from the averageand the associated standard deviation of a set of indications of the quantity, obtained independently. For Type Bevaluation, prior knowledge concerning the quantity is used to characterize the quantity by a PDF, from which a bestestimate of the quantity and the standard uncertainty associated with that estimate are determined. The GUM statesthat both types of evaluation are based on probability distributions [GUM:1995 3.3.4], and that both approaches employrecognized interpretations of probability [GUM:1995 3.3.5]. The GUM considers PDFs as underpinning uncertaintyevaluation: in the context of the law of propagation of uncertainty, it refers explicitly to input and output quantitiesas being describable or characterized by probability distributions [GUM:1995 G.6.6]. Also see 5.1.2.

5.11.3 The GUM uncertainty framework does not explicitly determine a PDF for the output quantity. However,the probability distribution used by that framework to characterize the output quantity is sometimes referred to inthis Supplement as “provided by” or “resulting from” the GUM uncertainty framework.

5.11.4 This Supplement attempts to provide an approach that is as consistent with the GUM as possible, especiallyrelating to the use of PDFs for all quantities, but departs from it in a clearly identified way where appropriate. Thesedepartures are:


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a) PDFs are explicitly assigned to all input quantities Xi (rather than associating standard uncertainties withestimates xi of Xi) based on information concerning these quantities. The classification into Type A and Type Bevaluations of uncertainty is not needed;

b) sensitivity coefficients [GUM:1995 5.1.3] are not an inherent part of the approach, and hence the calculation ornumerical approximation of the partial derivatives of the model with respect to the Xi is not required. Approxi-mations to sensitivity coefficients can, however, be provided that correspond to taking all higher-order terms inthe Taylor series expansion of the model into account (annex B);

c) a numerical representation of the distribution function for Y is obtained that is defined completely by the modeland the PDFs for the Xi, and not restricted to a Gaussian distribution or scaled and shifted t-distribution;

d) since the PDF for Y is not in general symmetric, a coverage interval for Y is not necessarily centred on theestimate of Y . Consideration therefore needs to be given to the choice of coverage interval corresponding to aspecified coverage probability.

5.11.5 Since the GUM uncertainty framework explicitly uses only best estimates xi and the associated uncertainties(and covariances and degrees of freedom where appropriate), it is restricted in the information it can provide about Y .Essentially it is limited to providing an estimate y of Y and the standard uncertainty u(y) associated with y, andperhaps the related (effective) degrees of freedom. y and u(y) will be valid for a model that is linear in X. Any otherinformation about Y , e.g. coverage intervals, is derived using additional assumptions, e.g. that the distribution for Yis Gaussian or a scaled and shifted t-distribution.

5.11.6 Some features of MCM are

a) reduction in the analysis effort required for complicated or non-linear models, especially since the partial deriva-tives of first- or higher-order used in providing sensitivity coefficients for the law of propagation of uncertaintyare not needed,

b) generally improved estimate of Y for non-linear models (cf. [GUM:1995 4.1.4]),

c) improved standard uncertainty associated with the estimate of Y for non-linear models, especially when the Xi

are assigned non-Gaussian (e.g. asymmetric) PDFs, without the need to provide derivatives of higher or-der [GUM:1995 5.1.2 note],

d) provision of a coverage interval corresponding to a stipulated coverage probability when the PDF for Y cannotadequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution, i.e. when thecentral limit theorem does not apply [GUM:1995 G.2.1, G.6.6]. Such an inadequate approximation can arisewhen (1) the PDF assigned to a dominant Xi is not a Gaussian distribution or a scaled and shifted t-distribution,(2) the model is non-linear, or (3) the approximation error incurred in using the Welch-Satterthwaite formula foreffective degrees of freedom is not negligible, and

e) a coverage factor [GUM:1995 2.3.6] is not required when determining a coverage interval.

6 Probability density functions for the input quantities

6.1 General

6.1.1 This clause gives guidance on the assignment, in some common circumstances, of PDFs to the input quanti-ties Xi in the formulation stage of uncertainty evaluation. Such an assignment can be based on Bayes’ theorem [20]or the principle of maximum entropy [8, 26, 51, 56].

NOTE In some circumstances, another approach for assigning a PDF may be useful. In any case, as in any scientific discipline,the reason for the decision should be recorded.

6.1.2 Generally, a joint PDF gX(ξ) is assigned to the input quantities X = (X1, . . . , XN )>. See 6.4.8.4 note 2.


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6.1.3 When the Xi are independent, PDFs gXi(ξi) are assigned individually based on an analysis of a series of

indications (Type A evaluation of uncertainty) or based on scientific judgement using information [50] such as historicaldata, calibrations, and expert judgement (Type B evaluation of uncertainty) [GUM:1995 3.3.5].

6.1.4 When some of the Xi are mutually independent, PDFs are assigned individually to them and a joint PDFto the remainder.

NOTE It may be possible to remove some or all dependencies by re-expressing relevant input quantities in terms of morefundamental independent input quantities on which the original input quantities depend [GUM:1995 F1.2.4, H.1.2]. Suchchanges can simplify both the application of the law of propagation of uncertainty and the propagation of distributions. Detailsand examples are available [15].

6.1.5 Information relevant to the assignment of PDFs to the Xi is contained in the GUM [GUM:1995 4.3].

6.1.6 Comprehensive guidance on the assignment of PDFs individually or jointly to the Xi is beyond the scopeof this Supplement. Such assigned PDFs encode the knowledge and expertise of the metrologist who formulates themodel and who is ultimately responsible for the quality of the final results.

6.1.7 A standard text on probability distributions is Evans, Hastings and Peacock [18].

6.2 Bayes’ theorem

6.2.1 Suppose that information about an input quantity X consists of a series of indications regarded as realiza-tions of independent, identically distributed random variables characterized by a specified PDF, but with unknownexpectation and variance. Bayes’ theorem can be used to calculate a PDF for X, where X is taken to be equal to theunknown average of these random variables. Calculation proceeds in two steps. First, a non-informative joint prior(pre-data) PDF is assigned to the unknown expectation and variance. Using Bayes’ theorem, this joint prior PDFis then updated, based on the information supplied by the series of indications, to yield a joint posterior (post-data) PDF for the two unknown parameters. The desired posterior PDF for the unknown average is then calculatedas a marginal PDF by integrating over the possible values of the unknown variance (see 6.4.9.2).

6.2.2 With the use of Bayes’ theorem, the updating is carried out by forming the product of a likelihood functionand the prior PDF [20]. The likelihood function, in the case of indications obtained independently, is the productof functions, one function for each indication and identical in form, e.g. to a Gaussian PDF. The posterior PDF isthen determined by integrating the product of prior PDF and likelihood over all possible values of the variance andnormalizing the resulting expression.

NOTE 1 In some cases (e.g. as in 6.4.11), the random variables, of which the indications are regarded as realizations, arecharacterized by a PDF with only one parameter. In such cases, a non-informative prior PDF is assigned to the unknownexpectation of the random variables, and the posterior distribution for X is given directly by Bayes’ theorem, without the needfor marginalisation.

NOTE 2 Bayes’ theorem can also be applied in other circumstances, e.g. when the expectation and standard deviation areunknown and equal.

6.3 Principle of maximum entropy

6.3.1 When using the principle of maximum entropy, introduced by Jaynes [25], a unique PDF is selected amongall possible PDFs having specified properties, e.g. specified central moments of different orders or specified intervalsfor which the PDF is non-zero. This method is particularly useful for assigning PDFs to quantities for which a seriesof indications is not available or to quantities that have not explicitly been measured at all.

6.3.2 In applying the principle of maximum entropy, to obtain a PDF gX(ξ) that adequately characterizes incom-plete knowledge about a quantity X according to the information available, the functional

S[g] = −∫

gX(ξ) ln gX(ξ) dξ,


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the “information entropy”, introduced by Shannon [48], is maximized under constraints given by the information.

6.4 Probability density function assignment for some common circumstances

6.4.1 General

Subclauses 6.4.2 to 6.4.11 provide assignments of PDFs to quantities based on various types of information regardingthose quantities. Given for each PDF gX(ξ) are

a) formulæ for the expectation and variance of X, and

b) the manner in which sampling from gX(ξ) can be undertaken.

Table 1 facilitates the use of these subclauses and also illustrates the corresponding PDFs.

NOTE These illustrations of the PDFs are not drawn to scale. The multivariate Gaussian PDF is not illustrated.

6.4.2 Rectangular distributions

6.4.2.1 If the only available information regarding a quantity X is a lower limit a and an upper limit b with a < b,then, according to the principle of maximum entropy, a rectangular distribution R(a, b) over the interval [a, b] wouldbe assigned to X.

6.4.2.2 The PDF for X is

gX(ξ) ={

1/(b− a), a ≤ ξ ≤ b,0, otherwise.

6.4.2.3 X has expectation and variance

E(X) =a + b

2, V (X) =

(b− a)2

12. (2)

6.4.2.4 To sample from R(a, b), make a draw r from the standard rectangular distribution R(0, 1) (see C.3.3), andform

ξ = a + (b− a)r.

6.4.3 Rectangular distributions with inexactly prescribed limits

6.4.3.1 A quantity X is known to lie between limits A and B with A < B, where the midpoint (A + B)/2 of theinterval defined by these limits is fixed and the length B−A of the interval is not known exactly. A is known to lie inthe interval a± d and B in b± d, where a, b and d, with d > 0 and a+ d < b− d, are specified. If no other informationis available concerning X, A and B, the principle of maximum entropy can be applied to assign to X a “curvilineartrapezoid” (a rectangular distribution with inexactly prescribed limits).


gX(ξ) =14d

ln[(w + d)/(x− ξ)], a− d ≤ ξ ≤ a + d,ln[(w + d)/(w − d)], a + d < ξ < b− d,ln[(w + d)/(ξ − x)], b− d ≤ ξ ≤ b + d,0, otherwise,

(3)

where x = (a + b)/2 and w = (b − a)/2 are, respectively, the midpoint and semi-width of the inter-val [a, b] [GUM:1995 4.3.9 note 2]. This PDF is trapezoidal-like, but has flanks that are not straight lines.


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Table 1 — Available information and the PDF assigned on the basis of that information (6.4.1, C.1.2)

Available information Assigned PDF and illustration (not to scale) Subclause

Lower and upper limits a, b Rectangular:R(a, b)

6.4.2

Inexact lower and upper limits a ± d,b± d

Curvilinear trapezoid:CTrap(a, b, d)

6.4.3

Sum of two quantities assigned rectan-gular distributions with lower and up-per limits a1, b1 and a2, b2

Trapezoidal:Trap(a, b, β) with a = a1 + a2,b = b1 + b2,β = |(b1 − a1)− (b2 − a2)|/(b− a)

6.4.4

Sum of two quantities assigned rectan-gular distributions with lower and up-per limits a1, b1 and a2, b2 and thesame semi-width (b1 − a1 = b2 − a2)

Triangular:T(a, b) with a = a1 + a2, b = b1 + b2

6.4.5

Sinusoidal cycling between lower andupper limits a, b

Arc sine (U-shaped):U(a, b)

6.4.6

Best estimate x and associated stan-dard uncertainty u(x)

Gaussian:N(x, u2(x))

6.4.7

Best estimate x of vector quantity andassociated uncertainty matrix Ux

Multivariate Gaussian:N(x, Ux)

6.4.8

Series of indications x1, . . . , xn sampledindependently from a quantity havinga Gaussian distribution, with unknownexpectation and unknown variance

Scaled and shifted t:

tn−1(x, s2/n) with x =n∑

i=1

xi/n,

s2 =n∑

i=1

(xi − x)2/(n− 1)

6.4.9.2

Best estimate x, expanded uncertaintyUp, coverage factor kp and effective de-grees of freedom νeff

Scaled and shifted t:tνeff (x, (Up/kp)2)

6.4.9.7

Best estimate x of non-negative quan-tity

Exponential:Ex(1/x)

6.4.10

Number q of objects counted Gamma:G(q + 1, 1)

6.4.11


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NOTE Formula (3) can be expressed as

gX(ξ) =1

4dmax

(ln

w + d

max(|ξ − x|, w − d), 0

)for computer implementation.


E(X) =a + b

2, V (X) =

(b− a)2

12+

d2

9. (4)

NOTE 1 The variance in expression (4) is always greater than the variance holding for exact limits in expression (2), i.e.when d = 0.

NOTE 2 The GUM treats the information about X in 6.4.3.1 by assigning a degrees of freedom to the standard uncertaintyassociated with the best estimate of X [GUM:1995 G.4.2].

6.4.3.4 To sample from CTrap(a, b, d), make two draws r1 and r2 independently from the standard rectangulardistribution R(0, 1) (see C.3.3), and form

as = (a− d) + 2dr1, bs = (a + b)− as,

and

ξ = as + (bs − as)r2.

NOTE as is a draw from the rectangular distribution with limits a ± d. bs is then formed to ensure that the midpoint of as

and bs is the prescribed value x = (a + b)/2.

EXAMPLE A certificate states that a voltage X lies in the interval 10.0 V ± 0.1 V. No other information is availableconcerning X, except that it is believed that the magnitude of the interval endpoints is the result of rounding correctly somenumerical value (see 3.20). On this basis, that numerical value lies between 0.05 V and 0.15 V, since the numerical value ofevery point in the interval (0.05, 0.15) rounded to one significant decimal digit is 0.1. The location of the interval can thereforebe regarded as fixed, whereas its width is inexact. The best estimate of X is x = 10.0 V and, using expression (4) basedon a = 9.9 V, b = 10.1 V and d = 0.05 V, the associated standard uncertainty u(x) is given by

u2(x) =(0.2)2

12+

(0.05)2

9= 0.003 6.

Hence u(x) = (0.003 6)1/2 = 0.060 V, which can be compared with 0.2/√

12 = 0.058 V in the case of exact limits, given byreplacing d by zero. The use of exact limits in this case gives a numerical value for u(x) that is 4 % smaller than that for inexactlimits. The relevance of such a difference needs to be considered in the context of the application.

6.4.4 Trapezoidal distributions

6.4.4.1 The assignment of a symmetric trapezoidal distribution to a quantity is discussed inthe GUM [GUM:1995 4.3.9]. Suppose a quantity X is defined as the sum of two independent quantities X1

and X2. Suppose, for i = 1 and i = 2, Xi is assigned a rectangular distribution R(ai, bi) with lower limit ai and upperlimit bi. Then the distribution for X is a symmetric trapezoidal distribution Trap(a, b, β) with lower limit a, upperlimit b, and a parameter β equal to the ratio of the semi-width of the top of the trapezoid to that of the base. Theparameters of this trapezoidal distribution are related to those of the rectangular distributions by

a = a1 + a2, b = b1 + b2, β =λ1

λ2, (5)

where

λ1 =|(b1 − a1)− (b2 − a2)|

2, λ2 =

b− a

2, (6)


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and

0 ≤ λ1 ≤ λ2.

6.4.4.2 The PDF for X (figure 5), obtained using convolution [42, p93], is

gX(ξ) =

(ξ − x + λ2)/(λ2

2 − λ21), x− λ2 ≤ ξ < x− λ1,

1/(λ1 + λ2), x− λ1 ≤ ξ ≤ x + λ1,(x + λ2 − ξ)/(λ2

2 − λ21), x + λ1 < ξ ≤ x + λ2,

0, otherwise,

(7)

where x = (a + b)/2.


gX(ξ) =1

λ1 + λ2min

(1

λ2 − λ1max (λ2 − |ξ − x|, 0) , 1

)for computer implementation.

��

�� @@

@@ -

6

a = x− λ2 x− λ1 x x + λ1 b = x + λ2

ξ

gX(ξ)

1λ1+λ2

Figure 5 — The trapezoidal PDF for X = X1 + X2, where the PDFs for X1 and X2 are rectangular (6.4.4.2)


E(X) =a + b

2, V (X) =

(b− a)2

24(1 + β2).

6.4.4.4 To sample from Trap(a, b, β), make two draws r1 and r2 independently from the standard rectangulardistribution R(0, 1) (see C.3.3), and form

ξ = a +b− a

2[(1 + β)r1 + (1− β)r2].

6.4.5 Triangular distributions

6.4.5.1 Suppose a quantity X is defined as the sum of two independent quantities, each assigned a rectangulardistribution (see 6.4.4), but with equal semi-widths, i.e. b1 − a1 = b2 − a2. It follows from expressions (5) and (6)that λ1 = 0 and β = 0. The distribution for X is the trapezoidal distribution Trap(a, b, 0), which reduces to the(symmetric) triangular distribution T(a, b) over the interval [a, b].


gX(ξ) =

(ξ − a)/w2, a ≤ ξ ≤ x,(b− ξ)/w2, x < ξ ≤ b,0, otherwise,

(8)

where x = (a + b)/2 and w = λ2 = (b− a)/2.


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gX(ξ) =2

b− amax

(1− 2|ξ − x|

b− a, 0

).

for computer implementation.


E(X) =a + b

2, V (X) =

(b− a)2

24.

6.4.5.4 To sample from T(a, b), make two draws r1 and r2 independently from the standard rectangular distribu-tion R(0, 1) (see C.3.3), and form

ξ = a +b− a

2(r1 + r2).

6.4.6 Arc sine (U-shaped) distributions

6.4.6.1 If a quantity X is known to cycle sinusoidally, with unknown phase Φ, between specified limits a and b,with a < b, then, according to the principle of maximum entropy, a rectangular distribution R(0, 2π) would be assignedto Φ. The distribution assigned to X is the arc sine distribution U(a, b) [18], given by the transformation

X =a + b

2+

b− a

2sinΦ,

where Φ has the rectangular distribution R(0, 2π).


gX(ξ) ={

(2/π)[(b− a)2 − (2ξ − a− b)2]−1/2, a < ξ < b,0, otherwise.

NOTE U(a, b) is related to the standard arc sine distribution U(0, 1) given by

gZ(z) =

{[z(1− z)]−1/2/π, 0 < z < 1,0, otherwise,

(9)

in the variable Z, through the linear transformation

X = a + (b− a)Z.

Z has expectation 1/2 and variance 1/8. The distribution (9) is termed the arc sine distribution, since the correspondingdistribution function is

GZ(z) =1

πarcsin (2z − 1) +

1

2.

It is a special case of the beta distribution with both parameters equal to one half.


E(X) =a + b

2, V (X) =

(b− a)2

8.

6.4.6.4 To sample from U(a, b), make a draw r from the standard rectangular distribution R(0, 1) (see C.3.3), andform

ξ =a + b

2+

b− a

2sin 2πr.


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6.4.7 Gaussian distributions

6.4.7.1 If a best estimate x and associated standard uncertainty u(x) are the only information available regardinga quantity X, then, according to the principle of maximum entropy, a Gaussian probability distribution N(x, u2(x))would be assigned to X.


gX(ξ) =1√

2πu(x)exp

(− (ξ − x)2

2u2(x)

). (10)


E(X) = x, V (X) = u2(x).

6.4.7.4 To sample from N(x, u2(x)), make a draw z from the standard Gaussian distribution N(0, 1) (see C.4), andform

ξ = x + u(x)z.

6.4.8 Multivariate Gaussian distributions

6.4.8.1 A comparable result to that in 6.4.7.1 holds for an N -dimensional quantity X = (X1, . . . , XN )>. If theonly information available is a best estimate x = (x1, . . . , xN )> of X and the associated (strictly) positive definiteuncertainty matrix

Ux =

u2(x1) u(x1, x2) · · · u(x1, xN )

u(x2, x1) u2(x2) · · · u(x2, xN )...

.... . .

...u(xN , x1) u(xN , x2) · · · u2(xN )

,

a multivariate Gaussian distribution N(x,Ux) would be assigned to X.

6.4.8.2 The joint PDF for X is

gX(ξ) =1

[(2π)N detUx]1/2exp

(−1

2(ξ − x)>Ux

−1(ξ − x))

. (11)

6.4.8.3 X has expectation and covariance matrix

E(X) = x, V (X) = Ux.

6.4.8.4 To sample from N(x,Ux), make N draws zi, i = 1, . . . , N , independently from the standard Gaussiandistribution N(0, 1) (see C.4), and form

ξ = x + R>z,

where z = (z1, . . . , zN )> and R is the upper triangular matrix given by the Cholesky decomposition Ux = R>R(see C.5).

NOTE 1 In place of the Cholesky decomposition Ux = R>R, any matrix factorization of this form can be used.

NOTE 2 The only joint PDFs considered explicitly in this Supplement are multivariate Gaussian, distributions commonlyused in practice. A numerical procedure for sampling from a multivariate Gaussian PDF is given above (and in C.5). If anothermultivariate PDF is to be used, a means for sampling from it would need to be provided.


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NOTE 3 The multivariate Gaussian PDF (11) reduces to the product of N univariate Gaussian PDFs when there are nocovariance effects. In that case

Ux = diag(u2(x1), . . . , u2(xN )),

whence

gX(ξ) =

N∏i=1

gXi(ξi),

with

gXi(ξi) =

1√2πu(xi)

exp

(− (ξi − xi)

2

2u2(xi)

).

6.4.9 t-distributions

6.4.9.1 t-distributions typically arise in two circumstances: the evaluation of a series of indications (see 6.4.9.2),and the interpretation of calibration certificates (see 6.4.9.7).

6.4.9.2 Suppose that a series of n indications x1, . . . , xn is available, regarded as being obtained independentlyfrom a quantity with unknown expectation µ0 and unknown variance σ2

0 having Gaussian distribution N(µ0, σ20).

The desired input quantity X is taken to be equal to µ0. Then, assigning a non-informative joint prior distributionto µ0 and σ2

0 , and using Bayes’ theorem, the marginal PDF for X is a scaled and shifted t-distribution tν(x, s2/n)with ν = n− 1 degrees of freedom, where

x =1n

n∑i=1

xi, s2 =1

n− 1

n∑i=1

(xi − x)2,

being, respectively, the average and variance of the indications [20].


gX(ξ) =Γ(n/2)

Γ((n− 1)/2)√

(n− 1)π× 1

s/√

n

(1 +

1n− 1

(ξ − x

s/√

n

)2)−n/2

, (12)

where

Γ(z) =∫ ∞

0

tz−1e−t dt, z > 0,

is the gamma function.


E(X) = x, V (X) =n− 1n− 3

s2

n,

where E(X) is defined only for n > 2 and V (X) only for n > 3. For n > 3, the best estimate of X and its associatedstandard uncertainty are therefore

x = x, u(x) =

√n− 1n− 3

s√n

. (13)

NOTE 1 In the GUM [GUM:1995 4.2], the standard uncertainty u(x) associated with the average of a series of n indicationsobtained independently is evaluated as u(x) = s/

√n, rather than from formula (13), and the associated degrees of freedom

ν = n− 1 is considered as a measure of the reliability of u(x). By extension, a degrees of freedom is associated with an uncertainty


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obtained from a Type B evaluation, based on subjective judgement of the reliability of the evaluation [GUM:1995 G.4.2](cf. 6.4.3.3 note 2). Degrees of freedom associated with the uncertainties u(xi) are necessary to obtain, by application of theWelch-Satterthwaite formula, the effective degrees of freedom νeff associated with the uncertainty u(y).

NOTE 2 In the Bayesian context of this Supplement, concepts such as the reliability, or the uncertainty, of an uncertainty arenot necessary. Accordingly, the degrees of freedom in a Type A evaluation of uncertainty is no longer viewed as a measure ofreliability, and the degrees of freedom in a Type B evaluation does not exist.

6.4.9.5 To sample from tν(x, s2/n), make a draw t from the central t-distribution tν with ν = n − 1 degrees offreedom [GUM:1995 G.3] (also see C.6), and form

ξ = x +s√n

t.

6.4.9.6 If instead of a standard deviation s calculated from a single series of indications, a pooled standarddeviation sp with νp degrees of freedom obtained from Q such sets,

s2p =

1νp

Q∑j=1

νjs2j , νp =

Q∑j=1

νj ,

is used, the degrees of freedom ν = n− 1 of the scaled and shifted t-distribution assigned to X should be replaced bythe degrees of freedom νp associated with the pooled standard deviation sp. As a consequence, formula (12) shouldbe replaced by

gX(ξ) =Γ((νp + 1)/2)Γ(νp/2)√νpπ

× 1sp/

√n

[1 +

1νp

(ξ − x

sp/√

n

)2]−(νp+1)/2

and expressions (13) by

x = x =1n

n∑i=1

xi, u(x) =√

νp

νp − 2sp√n

(νp ≥ 3).

6.4.9.7 If the source of information about a quantity X is a calibration certificate [GUM:1995 4.3.1] in which a bestestimate x, the expanded uncertainty Up, the coverage factor kp and the effective degrees of freedom νeff are stated,then a scaled and shifted t-distribution tν(x, (Up/kp)2) with ν = νeff degrees of freedom should be assigned to X.

6.4.9.8 If νeff is stated as infinite or not specified, in which case it would be taken as infinite in the absence ofother information, a Gaussian distribution N(x, (Up/kp)2) would be assigned to X (see 6.4.7.1).

NOTE This distribution is the limiting case of the scaled and shifted t-distribution tν(x, (Up/kp)2) as ν tends to infinity.

6.4.10 Exponential distributions

6.4.10.1 If the only available information regarding a non-negative quantity X is a best estimate x > 0 of X, then,according to the principle of maximum entropy, an exponential distribution Ex(1/x) would be assigned to X.


gX(ξ) ={

exp(−ξ/x)/x, ξ ≥ 0,0, otherwise.


E(X) = x, V (X) = x2.


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6.4.10.4 To sample from Ex(1/x), make a draw r from the standard rectangular distribution R(0, 1) (see C.3.3),and form

ξ = −x ln r.

NOTE Further information regarding the assignment of PDFs to non-negative quantities is available [14].

6.4.11 Gamma distributions

6.4.11.1 Suppose the quantity X is the average number of objects present in a sample of a fixed size (e.g. theaverage number of particles in an air sample taken from a clean room, or the average number of photons emitted by asource in a specified time interval). Suppose q is the number of objects counted in a sample of the specified size, and thecounted number is assumed to be a quantity with unknown expectation having a Poisson distribution. Then, accordingto Bayes’ theorem, after assigning a constant prior distribution to the expectation, a gamma distribution G(q + 1, 1)would be assigned to X.


gX(ξ) ={

ξq exp(−ξ)/q!, ξ ≥ 0,0, otherwise. (14)


E(X) = q + 1, V (X) = q + 1. (15)

6.4.11.4 To sample from G(q + 1, 1), make q + 1 draws ri, i = 1, . . . , q + 1, independently from the standardrectangular distribution R(0, 1) (see C.3.3), and form [18]

ξ = − lnq+1∏i=1

ri.

NOTE 1 If the counting is performed over several samples (according to the same Poisson distribution), and qi is the number ofobjects counted in the ith sample, of size Si, then the distribution for the average number of objects in a sample of size S =

∑iSi

is G(α, β) with α = 1 +∑

iqi and β = 1. Formulæ(14) and (15) apply with q =

∑i

qi.

NOTE 2 The gamma distribution is a generalization of the chi-squared distribution and is used to characterize informationassociated with variances.

NOTE 3 The particular gamma distribution in 6.4.11.4 is an Erlang distribution given by the sum of q + 1 exponentialdistributions with parameter 1 [18].

6.5 Probability distributions from previous uncertainty calculations

A previous uncertainty calculation may have provided a probability distribution for an output quantity that is tobecome an input quantity for a further uncertainty calculation. This probability distribution may be available ana-lytically in a recognized form, e.g. as a Gaussian PDF. It may be available as an approximation to the distributionfunction for a quantity obtained from a previous application of MCM, for example. Means for describing such adistribution function for a quantity is given in 7.5.1 and D.2.

7 Implementation of a Monte Carlo method

7.1 General

This clause gives information about the implementation of a Monte Carlo method for the propagation of distributions:see the procedure given in 5.9.6 and shown diagrammatically in figure 4.


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7.2 Number of Monte Carlo trials

7.2.1 A value of M , the number of Monte Carlo trials, i.e. the number of model evaluations to be made, needs tobe selected. It can be chosen a priori, in which case there will be no direct control over the quality of the numericalresults provided by MCM. The reason is that the number of trials needed to provide these results to a prescribednumerical tolerance will depend on the “shape” of the PDF for the output quantity and on the coverage probabilityrequired. Also, the calculations are stochastic in nature, being based on random sampling.

NOTE A value of M = 106 can often be expected to deliver a 95 % coverage interval for the output quantity such that thislength is correct to one or two significant decimal digits.

7.2.2 The choice of a value of M that is large compared with 1/(1−p), e.g. M at least 104 times greater than 1/(1−p),should be made. It can then be expected that G will provide a reasonable discrete representation of GY (η) in theregions near the endpoints of a 100p % coverage interval for Y .

7.2.3 Because there is no guarantee that this or any specific pre-assigned number will suffice, a procedure thatselects M adaptively, i.e. as the trials progress, can be used. Some guidance in this regard is available [2]. Subclause 7.9provides such a procedure, a property of which is that the number of trials taken is economically consistent with theexpectation of achieving a required numerical tolerance.

NOTE If the model is complicated, e.g. involving the solution of a finite-element model, because of large computing times itmay not be possible to use a sufficiently large value of M to obtain adequate distributional knowledge of the output quantity.In such a case an approximate approach would be to regard gY (η) as Gaussian (as in the GUM) and proceed as follows. Arelatively small value of M , 50 or 100, for example, would be used. The average and standard deviation of the resulting Mmodel values of Y would be taken as y and u(y), respectively. Given this information, a Gaussian PDF gY (η) = N(y, u2(y))would be assigned to characterize the knowledge of Y (see 6.4.7) and a desired coverage interval for Y calculated. Althoughthis use of a small value of M is inevitably less reliable than that of a large value in that it does not provide an approximationto the PDF for Y , it does take account of model non-linearity.

7.3 Sampling from probability distributions

In an implementation of MCM, M vectors xr, r = 1, . . . ,M (see 7.2), are drawn from the PDFs gXi(ξi) for the Ninput quantities Xi. Draws would be made from the joint (multivariate) PDF gX(ξ) if appropriate. Recommendationsconcerning the manner in which this sampling can be carried out are given in annex C for the commonest distributions,viz. the rectangular, Gaussian, t, and multivariate Gaussian. Also see 6.4. It is possible to draw at random from anyother distribution. See C.2. Some such distributions could be approximations to distributions based on Monte Carloresults from a previous uncertainty calculation (see 6.5, 7.5 and annex D).

NOTE For the results of MCM to be statistically valid, it is necessary that the pseudo-random number generators used to drawfrom the distributions required have appropriate properties. Some tests of randomness of the numbers produced by a generatorare indicated in C.3.2.

7.4 Evaluation of the model

7.4.1 The model is evaluated for each of the M draws from the PDFs for the N input quantities. Specifically,denote the M draws by x1, . . . ,xM , where the rth draw xr contains x1,r, . . . , xN,r, with xi,r a draw from the PDFfor Xi. Then, the model values are

yr = f(xr), r = 1, . . . ,M.

7.4.2 The necessary modifications are made to 7.4.1 if the Xi are not independent and hence a joint PDF is assignedto them.

NOTE Model and derivative evaluations are made when applying the law of propagation of uncertainty, using exact derivatives,at the best estimates of the input quantities. Model evaluations only are made when applying the law of propagation ofuncertainty when numerical (finite-difference) approximations to derivatives are used. These evaluations are made, if the GUMrecommendation [GUM:1995 5.1.3 note 2] is adopted, at the best estimates of the input quantities and at points perturbedby ± one standard uncertainty from each estimate in turn. With MCM, model evaluations are made in the neighbourhood of


JCGM 101:2008

these best estimates, viz. at points that can be expected to be up to several standard uncertainties away from these estimates.The fact that model evaluations are made at different points according to the approach used may raise issues regarding thenumerical procedure used to evaluate the model, e.g. ensuring its convergence (where iterative schemes are used) and numericalstability. The user should ensure that, where appropriate, the numerical methods used to evaluate f are valid for a sufficientlylarge region containing these best estimates. Only occasionally would it be expected that this aspect is critical.

7.5 Discrete representation of the distribution function for the output quantity

7.5.1 A discrete representation G of the distribution function GY (η) for the output quantity Y can be obtained asfollows:

a) sort the model values yr, r = 1, . . . ,M , provided by MCM into non-decreasing order. Denote the sorted modelvalues by y(r), r = 1, . . . ,M ;

b) if necessary, make minute numerical perturbations to any replicate model values y(r) in such a way that theresulting complete set of y(r), r = 1, . . . ,M , form a strictly increasing sequence (cf. condition b) in 5.10.1);

c) take G as the set y(r), r = 1, . . . ,M .

NOTE 1 With reference to step a), a sorting algorithm taking a number of operations proportional to M ln M should beused [47]. A naive algorithm would take a time proportional to M2, making the computation time unnecessarily long. See 7.8.

NOTE 2 In step a), the term “non-decreasing” rather than “increasing” is used because of possible equalities among the modelvalues yr.

NOTE 3 With reference to step b), making only minute perturbations will ensure that the statistical properties of the y(r)

are retained.

NOTE 4 In step b), it is exceedingly unlikely that perturbations are necessary, because of the very large number of distinctfloating-point numbers that can arise from model values generated from input quantities obtained as draws from random numbergenerators. A sound software implementation would make appropriate provision, however.

NOTE 5 With reference to step c), a variety of information can be deduced from G. In particular, information supplementaryto the expectation and standard deviation can be provided, such as measures of skewness and kurtosis, and other statistics suchas the mode and the median.

NOTE 6 If Y is to become an input quantity for a further uncertainty calculation, sampling from its probability distributionis readily carried out by drawing randomly from the y(r), r = 1, . . . , M , with equal probability (see 6.5).

7.5.2 The y(r) (or yr), when assembled into a histogram (with suitable cell widths) form a frequency distributionthat, when normalized to have unit area, provides an approximation to the PDF gY (η) for Y . Calculations are notgenerally carried out in terms of this histogram, the resolution of which depends on the choice of cell widths, but interms of G. The histogram can, however, be useful as an aid to understanding the nature of the PDF, e.g. the extentof its asymmetry. See, however, 7.8.3 note 1 regarding the use of a large numerical value of M .

7.5.3 A continuous approximation to GY (η) is sometimes useful. Annex D contains a means for obtaining such anapproximation.

7.6 Estimate of the output quantity and the associated standard uncertainty

The average

y =1M

M∑r=1

yr (16)


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and standard deviation u(y) determined from

u2(y) =1

M − 1

M∑r=1

(yr − y)2 (17)

are taken, respectively, as an estimate y of Y and the standard uncertainty u(y) associated with y.

NOTE 1 Formula (17) should be used rather than the mathematically equivalent formula

u2(y) =M

M − 1

(1

M

M∑r=1

y2r − y2

).

For the many circumstances in metrology in which u(y) is much smaller than |y| (in which case the yr have a number of leadingdecimal digits in common) the latter formula suffers numerically from subtractive cancellation (involving a mean square less asquared mean). This effect can be so severe that the resulting numerical value might have too few correct significant decimaldigits for the uncertainty evaluation to be valid [4].

NOTE 2 In some special circumstances, such as when one of the input quantities has been assigned a PDF based onthe t-distribution with fewer than three degrees of freedom, the expectation and standard deviation of Y , as described by

the PDF gY (η), might not exist. Formulæ(16) and (17) might not then provide meaningful results. A coverage intervalfor Y (see 7.7) can, however, be formed, since G is meaningful and can be determined.

NOTE 3 y will not in general agree with the model evaluated at the best estimates of the input quantities, since, for a non-linear model f(X), E(Y ) = E(f(X)) 6= f(E(X)) (cf. [GUM:1995 4.1.4]). Irrespective of whether f is linear or non-linear, inthe limit as M tends to infinity, y approaches E(f(X)) when E(f(X)) exists.

7.7 Coverage interval for the output quantity

7.7.1 A coverage interval for Y can be determined from the discrete representation G of GY (η) in an analogousmanner to that in 5.3.2 given GY (η).

7.7.2 Let q = pM , if pM is an integer. Otherwise, take q to be the integer part of pM + 1/2. Then [ylow, yhigh] isa 100p % coverage interval for Y , where, for any r = 1, . . . ,M −q, ylow = y(r) and yhigh = y(r+q). The probabilisticallysymmetric 100p % coverage interval is given by taking r = (M − q)/2, if (M − q)/2 is an integer, or the integerpart of (M − q + 1)/2, otherwise. The shortest 100p % coverage interval is given by determining r∗ such that,for r = 1, . . . ,M − q, y(r∗+q) − y(r∗) ≤ y(r+q) − y(r).

NOTE Because of the randomness in MCM, some of these M−q interval lengths will be shorter than they would be on average,and some longer. So, by choosing the least such length, (the approximation to) the shortest 100p % coverage interval tends tobe marginally shorter than that which would have been calculated from GY (η), with the consequence that the typical coverageprobability is less than 100p %. For large M , this coverage probability is negligibly less than 100p %.

EXAMPLE 105 numbers were drawn from a pseudo-random number generator for the rectangular distribution in the inter-val [0, 1], and the shortest 95 % coverage interval formed as above. This exercise was carried out 1 000 times. The averagecoverage probability was 94.92 % and the standard deviation of the 1 000 coverage probabilities 0.06 %.

7.8 Computation time

7.8.1 The computation time for MCM is dominated by that required for the following three steps:

a) make M draws from the PDF for each input quantity Xi (or the joint PDF for X);

b) make M corresponding evaluations of the model;

c) sort the resulting M model values into non-decreasing order.

7.8.2 The times taken in the three steps are directly proportional to (a) M , (b) M , and (c) M lnM (if an efficientsort algorithm [47] is used).


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7.8.3 If the model is simple and the input quantities are independent, the time in step c) can be expected todominate, and the overall time taken is typically a few seconds for M = 106 on a personal computer operating atseveral GHz. Otherwise, let T1 be the time taken to make one draw from the PDFs for the input quantities and T2

that to make one evaluation of the model. Then, the overall time can be taken as essentially M × (T1 + T2), which, ifthe model is complicated, is dominated by the term MT2.

NOTE 1 If the model is simple and M very large, e.g. 108 or 109, the sorting time may be excessive compared with the timetaken to make the M model evaluations. In such a case, calculations can instead be based on an approximation to gY (η) derivedfrom a suitable histogram of the yr.

NOTE 2 An indication of the computation time required for an application of MCM can be obtained as follows. Consider anartificial problem with a model consisting of the sum of five terms:

Y = cos X1 + sin X2 + tan−1 X3 + exp(X4) + X1/35 .

Assign a Gaussian PDF to each input quantity Xi. Make M = 106 Monte Carlo trials. The relative computation timesfor (a) generating 5M random Gaussian numbers, (b) forming M model values and (c) sorting the M model values wererespectively 20 %, 20 % and 60 %, with a total computation time of a few seconds on a personal computer operating atseveral GHz.

7.9 Adaptive Monte Carlo procedure

7.9.1 General

A basic implementation of an adaptive Monte Carlo procedure involves carrying out an increasing number of MonteCarlo trials until the various results of interest have stabilized in a statistical sense. A numerical result is deemedto have stabilized if twice the standard deviation associated with it is less than the numerical tolerance (see 7.9.2)associated with the standard uncertainty u(y).

7.9.2 Numerical tolerance associated with a numerical value

Let ndig denote the number of significant decimal digits regarded as meaningful in a numerical value z. The numericaltolerance δ associated with z is given as follows:

a) express z in the form c× 10`, where c is an ndig decimal digit integer and ` an integer;

b) take

δ =1210`. (18)

EXAMPLE 1 The estimate of the output quantity for a nominally 100 g measurement standard of mass [GUM:1995 7.2.2]is y = 100.021 47 g. The standard uncertainty u(y) = 0.000 35 g, both significant digits being regarded as meaningful.Thus, ndig = 2 and u(y) can be expressed as 35× 10−5 g, and so c = 35 and ` = −5. Take δ = 1

2× 10−5 g = 0.000 005 g.

EXAMPLE 2 As example 1 except that only one significant decimal digit in u(y) is regarded as meaningful. Thus, ndig = 1and u(y) = 0.000 4 g = 4× 10−4 g, giving c = 4 and ` = −4. Hence, δ = 1

2× 10−4 g = 0.000 05 g.

EXAMPLE 3 In a temperature measurement, u(y) = 2 K. Then, ndig = 1 and u(y) = 2 × 100 K, giving c = 2 and ` = 0.Thus, δ = 1

2× 100 K = 0.5 K.

7.9.3 Objective of adaptive procedure

The objective of the adaptive procedure given in 7.9.4 is to provide

a) an estimate y of Y ,

b) an associated standard uncertainty u(y), and


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c) the endpoints ylow and yhigh of a coverage interval for Y corresponding to a stipulated coverage probability

such that each of these four values can be expected to meet the numerical tolerance required.

NOTE 1 By its stochastic nature, the procedure cannot be guaranteed to provide such an interval.

NOTE 2 y and u(y) generally “converge” considerably faster than ylow and yhigh with respect to the number of Monte Carlotrials.

NOTE 3 Generally, the larger is the coverage probability, the larger is the number of Monte Carlo trials required to deter-mine ylow and yhigh for a given numerical tolerance.

7.9.4 Adaptive procedure

A practical approach, involving carrying out a sequence of applications of MCM, is as follows:

a) set ndig to an appropriate small positive integer (see 7.9.2);

b) set

M = max(J, 104),

where J is the smallest integer greater than or equal to 100/(1− p);

c) set h = 1, denoting the first application of MCM in the sequence;

d) carry out M Monte Carlo trials, as in 7.3 and 7.4;

e) use the M model values y1, . . . , yM so obtained to calculate, as in 7.5 to 7.7, y(h), u(y(h)), y(h)low and y

(h)high as an

estimate of Y , the associated standard uncertainty, and the left- and right-hand endpoints of a 100p % coverageinterval, respectively, i.e. for the hth member of the sequence;

f) if h = 1, increase h by one and return to step d);

g) calculate the standard deviation sy associated with the average of the estimates y(1), . . . , y(h) of Y , given by

s2y =

1h(h− 1)

h∑r=1

(y(r) − y)2,

where

y =1h

h∑r=1

y(r);

h) calculate the counterpart of this statistic for u(y), ylow and yhigh;

i) use all h×M model values available so far to form u(y);

j) calculate the numerical tolerance δ associated with u(y) as in 7.9.2;

k) if any of 2sy, 2su(y), 2sylow and 2syhigh exceeds δ, increase h by one and return to step d);

l) regard the overall computation as having stabilized, and use all h×M model values obtained to calculate y, u(y)and a 100p % coverage interval, as in 7.5 to 7.7.

NOTE 1 Normally ndig in step a) would be chosen to be 1 or 2.


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NOTE 2 The choice of M in step b) is arbitrary, but has been found suitable in practice.

NOTE 3 In step g), y can be regarded as a realization of a random variable with standard deviation sy.

NOTE 4 The standard deviations formed in steps g) and h) tend to reduce in a manner proportional to h−1/2 (cf. 5.9.6 note 2).

NOTE 5 In situations where a coverage interval is not required, the test for stabilization of the computation in step k) canbe based instead on 2sy and 2su(y) only.

NOTE 6 The factor 2 used in step k) is based on regarding the averages as realizations of Gaussian variables, and correspondsto a coverage probability of approximately 95 %.

NOTE 7 An alternative, non-adaptive approach for a 95 % probabilistically symmetric coverage interval, which can be obtainedusing the statistics of the binomial distribution [10], is as follows. Select M = 105 or M = 106. Form the interval [y(r), y(s)],

where, for M = 105, r = 2 420 and s = 97 581, or, for M = 106, r = 24 747 and s = 975 254. This interval is a 95 % statisticalcoverage interval at the level of confidence 0.99 [GUM:1995 C.2.30] [55], i.e. the coverage probability will be no less than 95 %in at least 99 % of uses of MCM. The average coverage probability of such an interval will be (s− r)/(M + 1), which is greaterthan 95 % by an amount that becomes smaller as M is increased, viz. 95.16 % for M = 105 and 95.05 % for M = 106. (Thereare other possibilities for r and s; they do not have to sum to M +1. A sufficient condition [10, section 2.6] is that s− r satisfies

M∑j=s−r

MCj pj(1− p)M−j < 1− 0.99,

where

MCj =M !

j!(M − j)!,

the best result being when this inequality is just satisfied.) These results can be extended to other coverage probabilities (andother choices of M).

8 Validation of results

8.1 Validation of the GUM uncertainty framework using a Monte Carlo method

8.1.1 The GUM uncertainty framework can be expected to work well in many circumstances. However, it is notalways straightforward to determine whether all the conditions for its application (see 5.7 and 5.8) hold. Indeed, thedegree of difficulty of doing so would typically be considerably greater than that required to apply MCM, assumingsuitable software were available [8]. Therefore, since these circumstances cannot readily be tested, any cases of doubtshould be validated. Since the domain of validity for MCM is broader than that for the GUM uncertainty framework,it is recommended that both the GUM uncertainty framework and MCM be applied and the results compared. Shouldthe comparison be favourable, the GUM uncertainty framework could be used on this occasion and for sufficientlysimilar problems in the future. Otherwise, consideration should be given to using MCM or another appropriateapproach instead.

8.1.2 Specifically, it is recommended that the two steps below and the following comparison process be carried out:

a) apply the GUM uncertainty framework (possibly with the law of propagation of uncertainty based on a higher-order Taylor series approximation) (see 5.6) to yield a 100p % coverage interval y ± Up for the output quantity,where p is the stipulated coverage probability;

b) apply the adaptive Monte Carlo procedure (see 7.9.4) to provide (approximations to) the standard uncertainty u(y)and the endpoints ylow and yhigh of the required (probabilistically symmetric or shortest) 100p % coverage intervalfor the output quantity. Also see 8.2.

8.1.3 A comparison procedure has the following objective: determine whether the coverage intervals obtained bythe GUM uncertainty framework and MCM agree to within a stipulated numerical tolerance. This numerical toleranceis assessed in terms of the endpoints of the coverage intervals and corresponds to that given by expressing the standard


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uncertainty u(y) to what is regarded as a meaningful number of significant decimal digits (cf. 7.9.2). The procedureis as follows:

a) form a numerical tolerance δ associated with u(y) as described in 7.9.2;

b) compare the coverage intervals obtained by the GUM uncertainty framework and MCM to determine whether therequired number of correct decimal digits in the coverage interval provided by the GUM uncertainty frameworkhas been obtained. Specifically, determine

dlow = |y − Up − ylow|, (19)dhigh = |y + Up − yhigh|, (20)

viz. the absolute differences of the respective endpoints of the two coverage intervals. Then, if both dlow and dhigh

are no larger than δ, the comparison is favourable and the GUM uncertainty framework has been validated inthis instance.

NOTE The choice of 100p % coverage interval will influence the comparison. Therefore, the validation applies for the specifiedcoverage probability p only.

8.2 Obtaining results from a Monte Carlo method for validation purposes

A sufficient number M of Monte Carlo trials (see 7.2) should be performed in obtaining MCM results for the validationpurposes of 8.1. Let ndig denote the number of significant decimal digits required in u(y) (see 7.9.1) when validatingthe GUM uncertainty framework using MCM. Let δ denote the numerical tolerance associated with u(y) (see 7.9.2).Then it is recommended that the adaptive Monte Carlo procedure (see 7.9.4) be used to provide MCM results to anumerical tolerance of δ/5. Such results can be obtained by replacing δ by δ/5 in step k) of that procedure.

NOTE It can be expected that the use of a numerical tolerance of δ/5 would require a value of M of the order of 25 timesthat for a numerical tolerance of δ. Such a value of M might present efficiency problems for some computers in operating withvector arrays of dimension M . In such a case, calculations can instead be based on an approximation to gY (η) derived from asuitable histogram of the yr, in which the cell frequencies in the histogram are updated as the Monte Carlo calculation proceeds.Cf. 7.8.3 note 1.

9 Examples

9.1 Illustrations of aspects of this Supplement

9.1.1 The examples given illustrate various aspects of this Supplement. They show the application of the GUMuncertainty framework with and without contributions derived from higher-order terms in the Taylor series approxi-mation of the model function. They also show the corresponding results provided by

a) MCM using pre-assigned numbers M of Monte Carlo trials,

b) the adaptive Monte Carlo procedure (see 7.9.4) in which M is determined automatically, or

c) both.

9.1.2 Some of the examples further show whether the MCM results provided in b) in 9.1.1 validate those provided bythe GUM uncertainty framework. A numerical tolerance δ (see 7.9.2) associated with u(y), with δ chosen appropriately,is used in comparing MCM and the GUM uncertainty framework. The Monte Carlo results provided in b) wereobtained using a numerical tolerance of δ/5 (see 8.2). In some instances, solutions are obtained analytically for furthercomparison.

9.1.3 Results are generally reported in the manner described in 5.5. However, more than the recommended oneor two significant decimal digits are often given to facilitate comparison of the results obtained from the variousapproaches.


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9.1.4 The Mersenne Twister generator [34] was used to generate pseudo-random numbers from a rectangular distri-bution (see C.3). It passes a comprehensive test for pseudo-random numbers drawn from a rectangular distribution [30](see C.3.2) and is available within MATLAB [36], the programming environment used to produce the results givenhere.

9.1.5 The first example (see 9.2) constitutes an additive model. It demonstrates that the results from MCM agreewith those from the application of the GUM uncertainty framework when the conditions hold for the latter (as in 5.7).The same model, but with different PDFs assigned to the input quantities, is also considered to demonstrate somedepartures when not all the conditions hold.

9.1.6 The second example (see 9.3) is a calibration problem from mass metrology. It demonstrates that the GUMuncertainty framework is valid in this instance only if the contributions derived from higher-order terms in the Taylorseries approximation of the model function are included.

9.1.7 The third example (see 9.4) is concerned with electrical measurement. It shows that the PDF for the outputquantity can be markedly asymmetric, and thus the GUM uncertainty framework can yield invalid results, even if allhigher-order terms are taken into account. Instances where the input quantities are independent and not independentare treated.

9.1.8 The fourth example (see 9.5) is that in the GUM concerned with gauge block calibration [GUM:1995 H.1].The information given there concerning the model input quantities is interpreted, PDFs accordingly assigned to thesequantities, and results from the GUM uncertainty framework and MCM obtained and compared. Moreover, thistreatment is applied both to the original model and the approximation made to it in the GUM.

9.2 Additive model

9.2.1 Formulation

This example considers the additive model

Y = X1 + X2 + X3 + X4, (21)

a special case of the generic linear model considered in the GUM, for three different sets of PDFs gXi(ξi) assigned to

the input quantities Xi, regarded as independent. The Xi and hence the output quantity Y have dimension 1. For thefirst set, each gXi

(ξi) is a standard Gaussian PDF (with Xi having expectation zero and standard deviation unity).For the second set, each gXi

(ξi) is a rectangular PDF, also with Xi having expectation zero and standard deviationunity. The third set is identical to the second except that the PDF for gX4(ξ4) has a standard deviation of ten.

NOTE Further information concerning additive models, such as the model (21), where the PDFs are Gaussian or rectangularor a combination of both, is available [13].

9.2.2 Normally distributed input quantities

9.2.2.1 Assign a standard Gaussian PDF to each Xi. The best estimates of the Xi are xi = 0, i = 1, 2, 3, 4, withassociated standard uncertainties u(xi) = 1.

9.2.2.2 The results obtained are summarized in the first five columns of table 2, with the results reported to threesignificant figures in order to facilitate their comparison (see 9.1.3).

NOTE The probabilistically symmetric 95 % coverage interval is determined, because the PDF for Y is known to be symmetricin this case, as it is for the other cases considered in this example.

9.2.2.3 The law of propagation of uncertainty [GUM:1995 5.1.2] gives the estimate y = 0.0 of Y and associatedstandard uncertainty u(y) = 2.0, using a numerical tolerance of two significant decimal digits for u(y) (δ = 0.05)(see 5.5). A probabilistically symmetric 95 % coverage interval for Y , based on a coverage factor of 1.96, is [−3.9, 3.9].


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9.2.2.4 The application of MCM (clause 7) with M = 105 trials gives y = 0.0, u(y) = 2.0 and the probabilisticallysymmetric 95 % coverage interval [−3.9, 3.9]. Two further applications of the method, with M = 106 trials, agreewith these results to within the numerical tolerance used. These two further applications (different random samplingsbeing made from the PDFs) were made to demonstrate the variation in the results obtained. The fourth and fifthnumerical values of M (1.23×106 and 1.02×106) are the numbers of trials for two applications of the adaptive MonteCarlo procedure (see 7.9) with the use of a numerical tolerance of δ/5 (see 8.2).

9.2.2.5 The PDF for Y obtained analytically is the Gaussian PDF with expectation zero and standard deviationtwo.

9.2.2.6 Figure 6 shows the (Gaussian) PDF for Y resulting from the GUM uncertainty framework. It also showsone of the approximations (scaled frequency distribution (histogram) of M = 106 model values of Y ) constituting thediscrete representation G (see 7.5) to this PDF provided by MCM. The endpoints of the probabilistically symmet-ric 95 % coverage interval provided by both methods are shown as vertical lines. The PDF and the approximationare visually indistinguishable, as are the respective coverage intervals. For this example, such agreement would be ex-pected (for a sufficiently large value of M), because all the conditions hold for the application of the GUM uncertaintyframework (see 5.7).

Table 2 — The application to the model (21), with each Xi assigned a standard Gaussian PDF, of (a) the

GUM uncertainty framework (GUF), (b) MCM, and (c) an analytical approach (9.2.2.2, 9.2.2.7, 9.2.3.4)

Method M y u(y) Probabilistically symmetric dlow dhigh GUF validated95 % coverage interval (δ = 0.05)?

GUF 0.00 2.00 [–3.92, 3.92]MCM 105 0.00 2.00 [–3.94, 3.92]MCM 106 0.00 2.00 [–3.92, 3.92]MCM 106 0.00 2.00 [–3.92, 3.92]

Adaptive MCM 1.23× 106 0.00 2.00 [–3.92, 3.93] 0.00 0.01 YesAdaptive MCM 1.02× 106 0.00 2.00 [–3.92, 3.92] 0.00 0.00 Yes

Analytical 0.00 2.00 [–3.92, 3.92]

Figure 6 — Approximations for the model (21), with each Xi assigned a standard Gaussian PDF, to the PDF

for Y provided by (a) the GUM uncertainty framework and (b) MCM (9.2.2.6, 9.2.3.3). “Unit” denotes any

unit

9.2.2.7 Columns 6 to 8 of table 2 also shows the results of applying the validation procedures of 8.1 and 8.2. Usingthe terminology of 7.9.2, ndig = 2, since two significant decimal digits in u(y) are sought. Hence, u(y) = 2.0 = 20×10−1,and so c = 20 and ` = −1. Thus, according to 7.9.2, the numerical tolerance is

δ =12× 10−1 = 0.05.


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The magnitudes dlow and dhigh of the endpoint differences (expressions (19) and (20)) are shown in table 2 for thetwo applications of the adaptive Monte Carlo procedure. Also shown is whether the GUM uncertainty framework hasbeen validated for δ = 0.05.

9.2.2.8 Figure 7 shows the length yhigh − ylow of the 95 % coverage interval for Y (see 7.7), as a function of theprobability at its left-hand endpoint, determined from G. As expected for a symmetric PDF, the interval takes itsshortest length when symmetrically located with respect to the expectation.

Figure 7 — The length of the 95 % coverage interval, as a function of the probability at its left-hand

endpoint, for the discrete representation G of the distribution function obtained by applying MCM to the

model (21) (9.2.2.8, 9.4.2.2.11)

9.2.2.9 Subclause 9.4 provides an example of an asymmetric PDF for which the shortest coverage interval differsappreciably from the probabilistically symmetric coverage interval.

9.2.3 Rectangularly distributed input quantities with the same width

9.2.3.1 Assign a rectangular PDF to each Xi, so that Xi has an expectation of zero and a standard deviation ofunity (in contrast to 9.2.2.1 where a Gaussian PDF is assigned). Again, the best estimates of the Xi are xi = 0,i = 1, 2, 3, 4, with associated standard uncertainties u(xi) = 1.

9.2.3.2 By following the analogous steps of 9.2.2.3 to 9.2.2.5, the results in table 3 were obtained. The analyticsolution for the endpoints of the probabilistically symmetric 95 % coverage interval, viz. ±2

√3[2− (3/5)1/4] ≈ ±3.88,

was obtained as described in annex E.

Table 3 — As table 2, but for rectangular PDFs, with the Xi having the same expectations and standard

deviations (9.2.3.2, 9.2.3.3, 9.2.3.4)


GUF 0.00 2.00 [–3.92, 3.92]MCM 105 0.00 2.01 [–3.90, 3.89]MCM 106 0.00 2.00 [–3.89, 3.88]MCM 106 0.00 2.00 [–3.88, 3.88]

Adaptive MCM 1.02× 106 0.00 2.00 [–3.88, 3.89] 0.04 0.03 YesAdaptive MCM 0.86× 106 0.00 2.00 [–3.87, 3.87] 0.05 0.05 No

Analytical 0.00 2.00 [–3.88, 3.88]

9.2.3.3 Figure 8 shows the counterpart of figure 6 in this case. By comparison with figure 6, some modest differencesbetween the approximations to the PDFs can be seen. The GUM uncertainty framework provides exactly the same PDFfor Y when the PDFs for the Xi are Gaussian or rectangular, because the expectations of these quantities are identical,


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as are the standard deviations, in the two cases. The PDF provided by MCM takes smaller values than those providedby the GUM uncertainty framework in the neighbourhood of the expectation and to a smaller extent towards thetails. It takes slightly greater values in the flanks. The endpoints of the coverage intervals provided are again almostvisually indistinguishable, but table 3 shows small differences.

9.2.3.4 The probabilistically symmetric 95 % coverage interval determined on the basis of the GUM un-certainty framework is in this case slightly more conservative than that obtained analytically. As for nor-mally distributed quantities, the validation procedure was applied (columns 6 to 8 of table 3). As before,ndig = 2, u(y) = 20× 10−1, c = 20, ` = −1 and δ = 0.05. The endpoint differences dlow and dhigh are larger thanfor the case of normally distributed quantities (table 2). For the first of the two applications of the adaptive MonteCarlo procedure, the GUM uncertainty framework is validated. For the second application, it is not validated, al-though dlow and dhigh for this application are close to the numerical tolerance δ = 0.05 (seen if more decimal digits thanin table 3 are considered). Different validation results such as these are an occasional consequence of the stochasticnature of the Monte Carlo method, especially in a case such as that here.

Figure 8 — The counterpart of figure 6 for quantities having the same expectations and standard deviations,

but rectangular PDFs (9.2.3.3)

9.2.4 Rectangularly distributed input quantities with different widths

9.2.4.1 Consider the example of 9.2.3, except that X4 has a standard deviation of ten rather than unity. Table 4contains the results obtained.

9.2.4.2 The numbers M of Monte Carlo trials taken by the adaptive procedure (0.03 × 106 and 0.08 × 106) aremuch smaller than they were for the two previous cases in this example. The main reason is that, in this case, δ = 0.5,the numerical tolerance resulting from requesting, as before, two significant decimal digits in u(y), is ten times theprevious value. Were the previous value to be used, M would be of the order of 100 times greater.

Table 4 — As table 3, except that the fourth input quantity has a standard deviation of ten rather than

unity, and no analytic solution is provided (9.2.4.1, 9.2.4.5)


GUF 0.0 10.1 [–19.9, 19.9]MCM 105 0.0 10.2 [–17.0, 17.0]MCM 106 0.0 10.2 [–17.0, 17.0]MCM 106 0.0 10.1 [–17.0, 17.0]

Adaptive MCM 0.03× 106 0.1 10.2 [–17.1, 17.1] 2.8 2.8 NoAdaptive MCM 0.08× 106 0.0 10.1 [–17.0, 17.0] 2.9 2.9 No


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9.2.4.3 Figure 9 shows the two approximations obtained to the PDF for Y . They differ appreciably. The dominanceof the PDF for X4 is evident. The PDF for Y resembles that for X4, but there is an effect in the flanks resulting fromthe PDFs for the other Xi.

9.2.4.4 Figure 9 also shows the endpoints of the probabilistically symmetric 95 % coverage interval for Y obtainedfrom these approximations. The inner pair of vertical lines indicates the endpoints of the probabilistically symmet-ric 95 % coverage interval determined by MCM. The outer pair results from the GUM uncertainty framework, with acoverage factor of k = 1.96.

Figure 9 — As figure 8, except that the fourth input quantity has a standard deviation of ten rather than

unity (9.2.4.3, 9.2.4.4)

9.2.4.5 The probabilistically symmetric 95 % coverage interval determined on the basis of the GUM uncertaintyframework in this case is more conservative than that obtained using MCM. Again, the validation procedure wasapplied (columns 6 to 8 of table 4). Now, ndig = 2, u(y) = 1.0×101 = 10×100, c = 10, ` = 0 and δ = 1/2×100 = 0.5.For the two applications of the adaptive Monte Carlo procedure, the GUM uncertainty framework is not validated.For a numerical tolerance of one significant decimal digit in u(y), i.e. ndig = 1, for which δ = 5, the validation statuswould be positive in both cases, the 95 % coverage intervals all being [−2× 101, 2× 101]. See 4.13.

NOTE The conditions for the central limit theorem to apply are not well met in this circumstance [GUM:1995 G.6.5], becauseof the dominating effect of the rectangular PDF for X4 (see 5.7.2). However, because these conditions are often in practiceassumed to hold, especially when using proprietary software for uncertainty evaluation (cf. 9.4.2.5 note 3), the characterizationof Y by a Gaussian PDF on the assumption of the applicability of this theorem is made in this subclause for comparisonpurposes.

9.3 Mass calibration

9.3.1 Formulation

9.3.1.1 Consider the calibration of a weight W of mass density ρW against a reference weight R of mass density ρR

having nominally the same mass, using a balance operating in air of mass density ρa [39]. Since ρW and ρR aregenerally different, it is necessary to account for buoyancy effects. Applying Archimedes’ principle, the model takesthe form

mW(1− ρa/ρW) = (mR + δmR)(1− ρa/ρR), (22)

where δmR is the mass of a small weight of density ρR added to R to balance it with W.


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9.3.1.2 It is usual to work in terms of conventional masses. The conventional mass mW,c of W is the mass of a(hypothetical) weight of density ρ0 = 8 000 kg/m3 that balances W in air at density ρa0 = 1.2 kg/m3. Thus,

mW(1− ρa0/ρW) = mW,c(1− ρa0/ρ0).

9.3.1.3 In terms of conventional masses mW,c, mR,c and δmR,c, the model (22) becomes

mW,c(1− ρa/ρW)(1− ρa0/ρW)−1 = (mR,c + δmR,c)(1− ρa/ρR)(1− ρa0/ρR)−1, (23)

from which, to an approximation adequate for most practical purposes,

mW,c = (mR,c + δmR,c)[1 + (ρa − ρa0)

(1

ρW− 1

ρR

)].

Let

δm = mW,c −mnom

be the deviation of mW,c from the nominal mass

mnom = 100 g.

The model used in this example is given by

δm = (mR,c + δmR,c)[1 + (ρa − ρa0)

(1

ρW− 1

ρR

)]−mnom. (24)

NOTE Applying the law of propagation of uncertainty to the “exact” model (23) is made difficult by the algebraic complexityof the partial derivatives. It is easier to apply MCM, because only model values need be formed.

9.3.1.4 The only information available concerning mR,c and δmR,c is a best estimate and an associated standarduncertainty for each of these quantities. Accordingly, following 6.4.7.1, a Gaussian distribution is assigned to each ofthese quantities, with these best estimates used as the expectations of the corresponding quantities and the associatedstandard uncertainties as the standard deviations. The only information available concerning ρa, ρW and ρR islower and upper limits for each of these quantities. Accordingly, following 6.4.2.1, a rectangular distribution isassigned to each of these quantities, with limits equal to the endpoints of the distribution. Table 5 summarizes theinput quantities and the PDFs assigned. In the table, a Gaussian distribution N(µ, σ2) is described in terms ofexpectation µ and standard deviation σ, and a rectangular distribution R(a, b) with endpoints a and b (a < b) interms of expectation (a + b)/2 and semi-width (b− a)/2.

NOTE The quantity ρa0 in the mass calibration model (24) is assigned the value 1.2 kg/m3 with no associated uncertainty.

Table 5 — The input quantities Xi and the PDFs assigned to them for the mass calibration model (24)

(9.3.1.4)

ParametersXi Distribution ——————————————————————————–

Expectation Standard Expectation Semi-widthµ deviation σ x = (a + b)/2 (b− a)/2

mR,c N(µ, σ2) 100 000.000 mg 0.050 mgδmR,c N(µ, σ2) 1.234 mg 0.020 mg

ρa R(a, b) 1.20 kg/m3 0.10 kg/m3

ρW R(a, b) 8× 103 kg/m3 1× 103 kg/m3

ρR R(a, b) 8.00× 103 kg/m3 0.05× 103 kg/m3

9.3.2 Propagation and summarizing

9.3.2.1 The GUM uncertainty framework and the adaptive Monte Carlo procedure (see 7.9) were each used toobtain an estimate δm of δm, the associated standard uncertainty u(δm), and the shortest 95 % coverage interval


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for δm. The results obtained are shown in table 6, in which GUF1 denotes the GUM uncertainty framework with first-order terms, MCM the adaptive Monte Carlo procedure, and GUF2 the GUM uncertainty framework with higher-orderterms.

9.3.2.2 0.72 × 106 trials were taken by the adaptive Monte Carlo procedure with the use of a numerical toler-ance of δ/5 (see 8.2) with δ set for the case where one significant decimal digit in u(δm) is regarded as meaningful(see 9.3.2.6).

9.3.2.3 Figure 10 shows the approximations to the PDF for δm obtained from the GUM uncertainty frameworkwith first-order terms and MCM. The continuous curve represents a Gaussian PDF with parameters given by the GUMuncertainty framework. The inner pair of (broken) vertical lines indicates the shortest 95 % coverage interval for δmbased on this PDF. The histogram is the scaled frequency distribution obtained using MCM as an approximation tothe PDF. The outer pair of (continuous) vertical lines indicates the shortest 95 % coverage interval for δm based onthe discrete representation of the distribution function determined as in 7.5.

Table 6 — Results of the calculation stage for the mass calibration model (24) (9.3.2.1, 9.3.2.6)

Method δm u(δm) Shortest 95 % dlow dhigh GUF validated/mg /mg coverage interval /mg /mg /mg (δ = 0.005)?

GUF1 1.234 0 0.053 9 [1.128 5, 1.339 5] 0.045 1 0.043 0 NoMCM 1.234 1 0.075 4 [1.083 4, 1.382 5]GUF2 1.234 0 0.075 0 [1.087 0, 1.381 0] 0.003 6 0.001 5 Yes

Figure 10 — Approximations to the PDF for the output quantity δm obtained using the GUM uncertainty

framework with first-order terms and MCM (9.3.2.3)

9.3.2.4 The results show that, although the GUM uncertainty framework (first order) and MCM give estimatesof δm in good agreement, the numerical values for the associated standard uncertainty are noticeably different. Thevalue (0.075 4 mg) of u(δm) returned by MCM is 40 % larger than that (0.053 9 mg) returned by the GUM uncertaintyframework (first order). The latter is thus optimistic in this respect. There is good agreement between u(δm)determined by MCM and that (0.075 0 mg) provided by the GUM uncertainty framework with higher-order terms.

9.3.2.5 Table 7 contains the partial derivatives of first order for the model (24) with respect to the input quantitiestogether with the sensitivity coefficients, viz. these derivatives evaluated at the best estimates of the input quantities.These derivatives indicate that, for the purposes of the GUM uncertainty framework with first-order terms, the modelfor this example can be considered as being replaced by the additive model

δm = mR,c + δmR,c −mnom.

MCM makes no such (implied) approximation to the model.


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Table 7 — Sensitivity coefficients for the mass calibration model (24) (9.3.2.5)

Xi Partial derivative Sensitivitycoefficient

mR,c 1 + (ρa − ρa0)(1/ρW − 1/ρR) 1δmR,c 1 + (ρa − ρa0)(1/ρW − 1/ρR) 1

ρa (mR,c + δmR,c)(1/ρW − 1/ρR) 0ρW −(mR,c + δmR,c)(ρa − ρa0)/ρ2

W 0ρR (mR,c + δmR,c)(ρa − ρa0)/ρ2

R 0

9.3.2.6 Table 6 also shows in the right-most three columns the results of applying the validation procedure of 8.1and 8.2 in the case where one significant decimal digit in u(δm) is regarded as meaningful. Using the terminologyof that subclause, ndig = 1, since a numerical tolerance of one significant decimal digit in u(δm) is required. Hence,u(δm) = 0.08 = 8× 10−2, and so the c in 7.9.2 equals 8 and ` = −2. Thus δ = 1/2 × 10−2 = 0.005. dlow and dhigh

denote the magnitudes of the endpoint differences (19) and (20), where y there corresponds to δm. Whether the resultswere validated to one significant decimal digit in u(δm) is indicated in the final column of the table. If only first-orderterms are accounted for, the application of the GUM uncertainty framework is not validated. If higher-order termsare accounted for [GUM:1995 5.1.2 note], the GUM uncertainty framework is validated. Thus, the non-linearity of themodel is such that accounting for first-order terms only is inadequate.

9.4 Comparison loss in microwave power meter calibration

9.4.1 Formulation

9.4.1.1 During the calibration of a microwave power meter, the power meter and a standard power meter areconnected in turn to a stable signal generator. The power absorbed by each meter will in general be different becausetheir complex input voltage reflection coefficients are not identical. The ratio Y of the power PM absorbed by themeter being calibrated and that, PS, by the standard meter is [43]

Y =PM

PS=

1− |ΓM|2

1− |ΓS|2× |1− ΓSΓG|2

|1− ΓMΓG|2, (25)

where ΓG is the voltage reflection coefficient of the signal generator, ΓM that of the meter being calibrated and ΓS

that of the standard meter. This power ratio is an instance of “comparison loss” [1, 28].

9.4.1.2 Consider the case where the standard and the signal generator are reflectionless, i.e. ΓS = ΓG = 0, andmeasured values are obtained of the real and imaginary parts X1 and X2 of ΓM = X1 + jX2, where j2 = −1.Since |ΓM|2 = X2

1 + X22 , formula (25) becomes

Y = 1−X21 −X2

2 . (26)

9.4.1.3 Given respectively are best estimates x1 and x2 of the quantities X1 and X2 from measurement and the as-sociated standard uncertainties u(x1) and u(x2). X1 and X2 are often not independent. Denote by u(x1, x2) the covari-ance associated with x1 and x2. Equivalently [GUM:1995 5.2.2], u(x1, x2) = r(x1, x2)u(x1)u(x2), where r = r(x1, x2)denotes the associated correlation coefficient [GUM:1995 5.2.2].

NOTE In practice the electrical engineer may sometimes have difficulty in quantifying the covariance. In such cases, theuncertainty evaluation can be repeated with different trial numerical values for the correlation coefficient in order to study itseffect. This example carries out calculations using a correlation coefficient of zero and of 0.9 (cf. 9.4.1.7).

9.4.1.4 On the basis of 6.4.8.1, X = (X1, X2)> is assigned a bivariate Gaussian PDF in X1 and X2, with expectationand covariance matrix[

x1

x2

],

[u2(x1) ru(x1)u(x2)

ru(x1)u(x2) u2(x2)

]. (27)


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9.4.1.5 Because the magnitudes of X1 and X2 in expression (26) are in practice small compared with unity, theresulting Y is close to unity. Results are accordingly expressed in terms of the quantity

δY = 1− Y = X21 + X2

2 , (28)

taken as the model of measurement. For physical reasons, 0 ≤ Y ≤ 1, and hence 0 ≤ δY ≤ 1.

9.4.1.6 The determination of an estimate δy of δY , the associated standard uncertainty u(δy), and a coverageinterval for δY will be considered for choices of x1, x2, u(x1), u(x2) and r(x1, x2). All quantities have dimension 1.

9.4.1.7 Six cases are considered, in all of which x2 is taken as zero and u(x1) = u(x2) = 0.005. The first three ofthese cases correspond to taking x1 = 0, 0.010, and 0.050, each with r(x1, x2) = 0. The other three cases correspondto taking the same x1, but with r(x1, x2) = 0.9. The various numerical values of x1 (comparable to those occurringin practice) are used to investigate the extent to which the results obtained using the considered approaches differ.

9.4.1.8 For the cases in which r = r(x1, x2) = 0, the covariance matrix given in formulæ (27) reducesto diag(u2(x1), u2(x2)) and the corresponding joint distribution for X1 and X2 to the product of two univariateGaussian distributions for Xi, for i = 1, 2, with expectation xi and standard deviation u(xi).

9.4.2 Propagation and summarizing: zero covariance

9.4.2.1 General

9.4.2.1.1 The evaluation of uncertainty is treated by applying the propagation of distributions

a) analytically (for purposes of comparison),

b) using the GUM uncertainty framework, and

c) using MCM.

NOTE These approaches do not constrain the PDF for δY to be no greater than unity. However, for sufficiently smalluncertainties u(x1) and u(x2), as here, the PDF for δY may adequately be approximated by a simpler PDF defined over allnon-negative values of δY . A rigorous treatment, using Bayesian inference [51], which applies regardless of the magnitudesof u(x1) and u(x2), is possible, but beyond the scope of this Supplement. Also see clause 1 note 2.

9.4.2.1.2 δy and u(δy) can generally be formed analytically as the expectation and standard deviation of δY , ascharacterized by the PDF for δY . See F.1. The PDF for δY can be formed analytically when x1 = 0 and, in particular,used to determine the endpoints of the shortest 95 % coverage interval in that case. See F.2.

9.4.2.1.3 The GUM uncertainty framework with first-order terms and with higher-order terms is applied for each ofthe three estimates x1 in the uncorrelated case. See F.3. An estimate δy of δY is formed in each case [GUM:1995 4.1.4]from

δy = x21 + x2

2.

9.4.2.1.4 MCM is applied in each case with M = 106 trials.

9.4.2.2 Input estimate x1 = 0

9.4.2.2.1 For the input estimate x1 = 0, higher-order terms must be used when applying the law of propagationof uncertainty, because the partial derivatives of δY with respect to X1 and X2, evaluated at X1 = x1 and X2 = x2,are identically zero when x1 = x2 = 0. Thus, if the law of propagation of uncertainty with first-order terms only wereapplied, the resulting standard uncertainty would incorrectly be computed as zero.

NOTE A similar difficulty would arise for x1 close to zero.


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9.4.2.2.2 Figure 11 shows the PDFs for δY determined by applying the propagation of distributions

a) analytically (the exponentially decreasing curve for δY ≥ 0 and zero elsewhere),

b) using the GUM uncertainty framework with higher-order terms in order to characterize the output quantity by aGaussian PDF (bell-shaped curve), and

c) using MCM (scaled frequency distribution).

Figure 11 — Results for the model of comparison loss in power meter calibration in the case x1 = x2 = 0,

with u(x1) = u(x2) = 0.005 and r(x1, x2) = 0 (9.4.2.2.2, 9.4.2.2.6, 9.4.2.2.9 and 9.4.2.2.11)

9.4.2.2.3 It is seen in the figure that the use of the GUM uncertainty framework with higher-order terms in orderto characterize the output quantity by a Gaussian distribution yields a PDF that is very different from the analyticsolution. The latter takes the form of a particular chi-squared distribution—the sum of squares of two standardGaussian variables (see F.2).

9.4.2.2.4 Since the partial derivatives of the model function (28) of order higher than two are all identically zero,the solution obtained essentially corresponds to taking all Taylor-series terms, i.e. the full non-linearity of the problem,into account. Thus, the particular Gaussian distribution so determined is the best that is possible using the GUMuncertainty framework to characterize the output quantity by such a distribution.

9.4.2.2.5 It can therefore be concluded that the reason for the departure from the analytic solution of the resultsfrom the approach based on the GUM uncertainty framework is that the output quantity is characterized by aGaussian PDF. No Gaussian PDF, however it is obtained, could adequately represent the analytic solution in thiscase.

9.4.2.2.6 It is also seen in figure 11 that the PDF provided by MCM is consistent with the analytic solution.

9.4.2.2.7 The estimates δy determined as the expectation of δY described by the PDFs obtained

a) analytically,

b) using the GUM uncertainty framework, and

c) applying MCM

are given in columns 2 to 4 of the row corresponding to x1 = 0.000 in table 8. Columns 5 to 8 contain the correspond-ing u(δy), with those obtained using the GUM uncertainty framework with first-order terms (G1) and higher-orderterms (G2).


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Table 8 — Comparison loss results, for input estimates with associated zero covariance, obtained

analytically (A), and using the GUM uncertainty framework with first-order terms (G1) and higher-order

terms (G2) and MCM (M) (9.4.2.2.7, 9.4.2.2.10, 9.4.2.3.4, 9.4.2.4.2)

Estimate Standard uncertainty Shortest 95 % coverage interval forx1 δy /10−6 u(δy) /10−6 δY /10−6

A G M A G1 G2 M A G1 G2 M0.000 50 0 50 50 0 50 50 [0, 150] [0, 0] [–98, 98] [0, 150]0.010 150 100 150 112 100 112 112 — [–96, 296] [–119, 319] [0, 367]0.050 2 550 2 500 2 551 502 500 502 502 — [1 520, 3 480] [1 515, 3 485] [1 590, 3 543]

9.4.2.2.8 The estimate δy = 0 obtained by evaluating the model at the input estimates is invalid: the correct(analytic) PDF for δY is identically zero for δY < 0; this estimate lies on the boundary of the non-zero part ofthat function. The estimate provided by MCM agrees with that obtained analytically. The law of propagation ofuncertainty based on first-order terms gives the wrong, zero, value for u(δy) already noted. The value (50 × 10−6)from the law of propagation of uncertainty based on higher-order terms agrees with that obtained analytically andby MCM.

NOTE When MCM was repeated several times the results obtained were scattered about 50 × 10−6. When it was repeateda number of times with a larger numerical value of M the results were again scattered about 50 × 10−6, but with a reduceddispersion. Such dispersion effects are expected, and were observed for the other Monte Carlo calculations made. Reportingthe results to greater numbers of significant decimal digits would be necessary to see the actual numerical differences.

9.4.2.2.9 Figure 11 also shows the shortest 95 % coverage intervals for the corresponding approximations to thedistribution function for δY . The 95 % coverage interval, indicated by dotted vertical lines, as provided by the GUMuncertainty framework is infeasible: it is symmetric about δY = 0 and therefore erroneously implies there is a 50 %probability that δY is negative. The continuous vertical lines are the endpoints of the shortest 95 % coverage intervalderived from the analytic solution, as described in F.2. The endpoints of the shortest 95 % coverage interval determinedusing MCM are indistinguishable to graphical accuracy from those for the analytic solution.

9.4.2.2.10 The endpoints of the shortest coverage intervals relating to the standard uncertainties in columns 5 to 8of the row corresponding to x1 = 0.000 in table 8 are given in columns 9 to 12 of that table.

9.4.2.2.11 Figure 12 shows the length of the 95 % coverage interval (see 7.7), as a function of the probability valueat its left-hand endpoint, for the approximation to the PDF provided by MCM shown in figure 11. The 95 % coverageinterval does not take its shortest length when symmetrically located with respect to the expectation in this case.Indeed, the shortest 95 % coverage interval is as far-removed as possible from a probabilistically symmetric coverageinterval, the left and right tail probabilities being 0 % and 5 %, respectively, as opposed to 2.5 % and 2.5 %. Thisfigure can be compared with that (figure 7) for the additive model of 9.2, for which the PDF for Y is symmetric aboutits expectation.

9.4.2.3 Input estimate x1 = 0.010

9.4.2.3.1 For the input estimate x1 = 0.010, with correlation coefficient r(x1, x2) = 0, figure 13 shows the PDFsobtained using the GUM uncertainty framework with first-order terms only and with higher-order terms, and us-ing MCM.

9.4.2.3.2 The PDF provided by MCM exhibits a modest left-hand flank, although it is truncated at zero, thesmallest possible numerical value of δY . Further, compared with the results for x1 = 0, it is closer in form to theGaussian PDFs provided by the GUM uncertainty framework. These Gaussian PDFs are in turn reasonably close toeach other, δY having expectation 1.0× 10−4 and standard deviations 1.0× 10−4 and 1.1× 10−4, respectively.

9.4.2.3.3 Figure 13 also shows the endpoints of the shortest 95 % coverage intervals obtained by the three ap-proaches. The continuous vertical lines denote the endpoints of the interval provided by MCM, the broken verticallines those resulting from the GUM uncertainty framework with first-order terms, and the dotted vertical lines fromthe GUM uncertainty framework with higher-order terms. The intervals provided by the GUM uncertainty frameworkare shifted to the left compared with the shortest 95 % coverage interval for MCM. As a consequence, they again


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Figure 12 — The length of the 95 % coverage interval, as a function of the probability value at its left-hand

endpoint, for the approximation to the distribution function obtained by applying MCM to the

model (28) (9.4.2.2.11)

Figure 13 — As figure 11 except that x1 = 0.010, and the PDFs resulting from the GUM uncertainty

framework with first-order (higher-peaked curve) and with higher-order terms (lower-peaked curve)

(9.4.2.3.1, 9.4.2.3.3, 9.4.2.4.1, 9.4.3.3)


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include infeasible values of δY . The shift is about 70 % of the standard uncertainty. The interval provided by MCMhas its left-hand endpoint at zero, the smallest feasible value.

9.4.2.3.4 The corresponding results are given in the penultimate row of table 8.

9.4.2.4 Input estimate x1 = 0.050

9.4.2.4.1 Figure 14 is similar to figure 13, but for x1 = 0.050. Now, the PDFs provided by both variants ofthe GUM uncertainty framework are virtually indistinguishable from each other. Further, they are now much closerto the approximation to the PDF provided by MCM. That PDF exhibits a slight skewness, as evidenced in the tailregions. The coverage intervals provided by the two variants of the GUM uncertainty framework are visually almostidentical, but still shifted from those for MCM. The shift is now about 10 % of the standard uncertainty. The intervalsprovided by the GUM uncertainty framework are now feasible.

Figure 14 — As figure 13 except that x1 = 0.050 (9.4.2.4.1, 9.4.3.3)

9.4.2.4.2 The corresponding results are given in the final row of table 8.

9.4.2.5 Discussion

As x1 becomes increasingly removed from zero, the results given by the GUM uncertainty framework, with first-orderand with higher-order terms, and those for MCM become closer to each other.

NOTE 1 The numerical values x1 = x2 = 0 lie in the centre of the region of interest to the electrical engineer, correspondingto the so-called “matched” condition for the power meter being calibrated, and thus in no sense constitute an extreme case.

NOTE 2 Because of the symmetry of the model in X1 and X2, exactly the same effect would occur were x2 used in placeof x1.

NOTE 3 One reason why the GUM uncertainty framework with first-order terms (only) might be used in practice is thatsoftware for its implementation is readily available: results obtained from it might sometimes be accepted without question. Forthe case where x1 = x2 = 0 (figure 11), the danger would be apparent because the standard uncertainty u(δy) was computedas zero, and consequently any coverage interval for δY would be of zero length for any coverage probability. For x1 6= 0(or x2 6= 0), u(δy) and the length of the coverage interval for δY are both non-zero, so no such warning would be availablewithout prior knowledge of likely values for u(δy) and this length. Thus, a danger in implementing software based on the GUMuncertainty framework for these calculations is that checks of the software for x1 or x2 sufficiently far from zero would notindicate such problems, although, when used subsequently in practice for small values of x1 or x2, the results would be invalid,but conceivably unwittingly accepted.


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9.4.3 Propagation and summarizing: non-zero covariance

9.4.3.1 General

9.4.3.1.1 The three approaches used in the cases where the Xi are uncorrelated (see 9.4.2) are now applied forthe three cases in which they are correlated, with r(x1, x2) = 0.9. However, the GUM uncertainty framework withfirst-order terms only is used. Unlike the cases where the Xi are uncorrelated, the GUM uncertainty framework withhigher-order terms is not applied, no counterpart being provided in the GUM for the formula containing higher-orderterms when the xi have associated non-zero covariances (see 5.8). Other aspects match those in 9.4.2.

9.4.3.1.2 For the GUM uncertainty framework with first-order terms, u(δy) is evaluated as described in F.3.2.Expression (F.7) in that subclause gives, for x2 = 0,

u2(δy) = 4x21u

2(x1).

Consequently, u(δy) does not depend on r(x1, x2) and the GUM uncertainty framework with first-order terms givesidentical results to those presented in 9.4.2. In particular, for the case x1 = 0, u(δy) is (incorrectly) computed as zero,as in 9.4.2.2.1.

9.4.3.1.3 MCM was implemented by sampling randomly from X characterized by a bivariate Gaussian PDF withthe given expectation and covariance matrix (expressions (27)). The procedure in C.5 was used.

NOTE Apart from the requirement to draw from a multivariate distribution, the implementation of MCM for input quantitiesthat are correlated is no more complicated than when the input quantities are uncorrelated.

9.4.3.2 Input estimates x1 = 0, 0.010, and 0.050

9.4.3.2.1 Table 9 contains the results obtained. Those from MCM indicate that although δy is unaffected by thecorrelation between the Xi, u(δy) is so influenced, more so for small x1. The 95 % coverage intervals are influencedaccordingly.

Table 9 — Comparison loss results, for input estimates with associated non-zero covariance (r(x1, x2) = 0.9),

obtained analytically, and using the GUM uncertainty framework (GUF) and MCM (9.4.3.2.1)

Estimate Standard uncertainty Shortest 95 % coverage interval forx1 δy /10−6 u(δy) /10−6 δY /10−6

Analytical GUF MCM Analytical GUF MCM Analytical GUF MCM0.000 50 0 50 67 0 67 — [0, 0] [0, 185]0.010 150 100 150 121 100 121 — [−96, 296] [13, 398]0.050 2 550 2 500 2 551 505 500 504 — [1 520, 3 480] [1 628, 3 555]

9.4.3.2.2 Figures 15 and 16 show the PDFs provided by the GUM uncertainty framework with first-order terms(bell-shaped curves) and MCM (scaled frequency distributions) in the cases x1 = 0.010 and x1 = 0.050, respectively.The endpoints of the shortest 95 % coverage interval provided by the two approaches are also shown, as broken verticallines for the GUM uncertainty framework and continuous vertical lines for MCM.

NOTE Strictly, the conditions under which δY can be characterized by a Gaussian PDF do not hold following an applicationof the GUM uncertainty framework in this circumstance (see 5.8) [GUM:1995 G.6.6]. However, this PDF and the endpoints ofthe corresponding 95 % coverage interval are shown because such a characterization is commonly used.

9.4.3.3 Discussion

In the case x1 = 0.010 (figure 15), the effect of the correlation has been to change noticeably the results returnedby MCM (compare with figure 13). Not only has the shape of (the approximation to) the PDF changed, but thecorresponding coverage interval no longer has its left-hand endpoint at zero. In the case x1 = 0.050 (figure 16), thedifferences between the results for the cases where the input quantities are uncorrelated and correlated (compare withfigure 14) are less obvious.


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Figure 15 — Results for the model of comparison loss in power meter calibration in the case x1 = 0.010, x2 = 0,

with u(x1) = u(x2) = 0.005 and r(x1, x2) = 0.9 (9.4.3.2.2, 9.4.3.3)

Figure 16 — As figure 15 except that x1 = 0.050 (9.4.3.2.2, 9.4.3.3)


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9.5 Gauge block calibration

9.5.1 Formulation: model

9.5.1.1 The length of a nominally 50 mm gauge block is determined by comparing it with a known referencestandard of the same nominal length. The direct output of the comparison of the two gauge blocks is the difference din their lengths given by

d = L(1 + αθ)− Ls(1 + αsθs), (29)

where L is the length at 20 ◦C of the gauge block being calibrated, Ls is the length of the reference standard at 20 ◦Cas given in its calibration certificate, α and αs are the coefficients of thermal expansion, respectively, of the gaugebeing calibrated and the reference standard, and θ and θs are the deviations in temperature from the 20 ◦C referencetemperature, respectively, of the gauge block being calibrated and the reference standard.

NOTE 1 The GUM refers to a gauge block as an end gauge.

NOTE 2 The symbol L for the length of a gauge block is used in this Supplement in place of the symbol ` used in the GUMfor that quantity.

9.5.1.2 From expression (29), the output quantity L is given by

L =Ls(1 + αsθs) + d

1 + αθ, (30)

from which, to an approximation adequate for most practical purposes,

L = Ls + d + Ls(αsθs − αθ). (31)

If the difference in temperature between the gauge block being calibrated and the reference standard is writtenas δθ = θ− θs, and the difference in their thermal expansion coefficients as δα = α−αs, models (30) and (31) become,respectively,

L =Ls[1 + αs(θ − δθ)] + d

1 + (αs + δα)θ(32)

and

L = Ls + d− Ls(θδα + αsδθ). (33)

9.5.1.3 The difference d in the lengths of the gauge block being calibrated and the reference standard is determinedas the average of a series of five indications, obtained independently, of the difference using a calibrated comparator.d can be expressed as

d = D + d1 + d2, (34)

where D is a quantity of which the average of the five indications is a realization, and d1 and d2 are quantitiesdescribing, respectively, the random and systematic effects associated with using the comparator.

9.5.1.4 The quantity θ, representing deviation of the temperature from 20 ◦C of the gauge block being calibrated,can be expressed as

θ = θ0 + ∆, (35)

where θ0 is a quantity representing the average temperature deviation of the gauge block from 20 ◦C and ∆ a quantitydescribing a cyclic variation of the temperature deviation from θ0.


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9.5.1.5 Substituting expressions (34) and (35) into expressions (32) and (33), and working with the quantity δLrepresenting the deviation of L from the nominal length

Lnom = 50 mm

of the gauge block, gives

δL =Ls[1 + αs(θ0 + ∆− δθ)] + D + d1 + d2

1 + (αs + δα)(θ0 + ∆)− Lnom (36)

and

δL = Ls + D + d1 + d2 − Ls[δα(θ0 + ∆) + αsδθ]− Lnom (37)

as models for the measurement problem.

9.5.1.6 The treatment here of the measurement problem is in terms of the models (36) and (37) with outputquantity δL and input quantities Ls, D, d1, d2, αs, θ0, ∆, δα and δθ. It differs from that given in GUM example H.1in that in the GUM the models (34) and (35) above are treated as sub-models to models (32) and (33), i.e. the GUMuncertainty framework is applied to each model (34) and (35), with the results obtained used to provide informationabout the input quantities d and θ in models (32) and (33). The treatment here avoids having to use the resultsobtained from MCM applied to the sub-models (34) and (35) to provide information about the distributions for theinput quantities d and θ in expressions (32) and (33).

9.5.2 Formulation: assignment of PDFs

9.5.2.1 General

In the following subclauses the available information about each input quantity in the models (36) and (37) is provided.This information is extracted from the description given in the GUM, and for each item of information the GUMsubclause from which the item is extracted is identified. Also provided is an interpretation of the information in termsof an assignment of a distribution to the quantity. Table 10 summarizes the assignments made.

Table 10 — PDFs assigned to input quantities for the gauge block models (36) and (37) on the basis of

available information (9.5.2.1). Table 1 provides general information concerning these PDFs

ParametersQuan- PDF ——————————————————————————————————–tity µ σ ν a b dLs tν(µ, σ2) 50 000 623 nm 25 nm 18D tν(µ, σ2) 215 nm 6 nm 24d1 tν(µ, σ2) 0 nm 4 nm 5d2 tν(µ, σ2) 0 nm 7 nm 8αs R(a, b) 9.5× 10−6 ◦C−1 13.5× 10−6 ◦C−1

θ0 N(µ, σ2) −0.1 ◦C 0.2 ◦C∆ U(a, b) −0.5 ◦C 0.5 ◦Cδα CTrap(a, b, d) −1.0× 10−6 ◦C−1 1.0× 10−6 ◦C−1 0.1× 10−6 ◦C−1

δθ CTrap(a, b, d) −0.050 ◦C 0.050 ◦C 0.025 ◦C

9.5.2.2 Length Ls of the reference standard

9.5.2.2.1 Information

The calibration certificate for the reference standard gives Ls = 50.000 623 mm as its length at 20 ◦C [GUM:1995 H.1.5].It gives Up = 0.075 µm as the expanded uncertainty of the reference standard and states that it was obtained using acoverage factor of kp = 3 [GUM:1995 H.1.3.1]. The certificate states that the effective degrees of freedom associatedwith the combined standard uncertainty, from which the quoted expanded uncertainty was obtained, is νeff(u(Ls)) = 18[GUM:1995 H.1.6].


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9.5.2.2.2 Interpretation

Assign a scaled and shifted t-distribution tν(µ, σ2) (see 6.4.9.7) to Ls, with

µ = 50 000 623 nm, σ =Up

kp=

753

nm = 25 nm, ν = 18.

9.5.2.3 Average length difference D


The average D of the five indications of the difference in lengths between the gauge block being calibrated and thereference standard is 215 nm [GUM:1995 H.1.5]. The pooled experimental standard deviation characterizing thecomparison of L and Ls was determined from 25 indications, obtained independently, of the difference in lengths oftwo standard gauge blocks, and equalled 13 nm [GUM:1995 H.1.3.2].


Assign a scaled and shifted t-distribution tν(µ, σ2) (see 6.4.9.2 and 6.4.9.6) to D, with

µ = 215 nm, σ =13√

5nm = 6 nm, ν = 24.

9.5.2.4 Random effect d1 of comparator


According to the calibration certificate of the comparator used to compare L with Ls, the associated uncertaintydue to random effects is 0.01 µm for a coverage probability of 95 % and is obtained from six indications, obtainedindependently [GUM:1995 H.1.3.2].


Assign a scaled and shifted t-distribution tν(µ, σ2) (see 6.4.9.7) to d1, with

µ = 0 nm, σ =U0.95

k0.95=

102.57

nm = 4 nm, ν = 5.

Here, k0.95 is obtained from table G.2 of the GUM with ν = 5 degrees of freedom and p = 0.95.

9.5.2.5 Systematic effect d2 of comparator


The uncertainty of the comparator due to systematic effects is given in the certificate as 0.02 µm at the “three sigmalevel” [GUM:1995 H.1.3.2]. This uncertainty may be assumed to be reliable to 25 %, and thus the degrees of freedomis νeff(u(d2)) = 8 [GUM:1995 H.1.6].


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Assign a scaled and shifted t-distribution tν(µ, σ2) (see 6.4.9.7) to d2, with

µ = 0 nm, σ =Up

kp=

203

nm = 7 nm, ν = 8.

9.5.2.6 Thermal expansion coefficient αs


The coefficient of thermal expansion of the reference standard is given as αs = 11.5× 10−6 ◦C−1 with possible valuesof this quantity represented by a rectangular distribution with limits ±2× 10−6 ◦C−1 [GUM:1995 H.1.3.3].


Assign a rectangular distribution R(a, b) (see 6.4.2) to αs, with limits

a = 9.5× 10−6 ◦C−1, b = 13.5× 10−6 ◦C−1.

NOTE There is no information about the reliability of the limits and so a rectangular distribution with exactly known limitsis assigned. Such information may have been omitted from the description in the GUM because the corresponding sensitivitycoefficient is zero, and so the quantity makes no contribution in an application of the GUM uncertainty framework based onfirst-order terms only.

9.5.2.7 Average temperature deviation θ0


The temperature of the test bed is reported as (19.9± 0.5) ◦C. The average temperature deviation θ0 = −0.1 ◦C isreported as having an associated standard uncertainty due to the uncertainty associated with the average temperatureof the test bed of u(θ0) = 0.2 ◦C [GUM:1995 H.1.3.4].


Assign a Gaussian distribution N(µ, σ2) (see 6.4.7) to θ0, with

µ = −0.1 ◦C, σ = 0.2 ◦C.

NOTE There is no information about the source of the evaluation of the uncertainty and so a Gaussian distribution is assigned.Also see 9.5.2.6.2 note, regarding such information.

9.5.2.8 Effect ∆ of cyclic temperature variation


The temperature of the test bed is reported as (19.9± 0.5) ◦C. The stated maximum offset of 0.5 ◦C for ∆ is saidto represent the amplitude of an approximately cyclical variation of temperature under a thermostatic system. Thecyclic variation of temperature results in a U-shaped (arc sine) distribution [GUM:1995 H.1.3.4].


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Assign an arc sine distribution U(a, b) (see 6.4.6) to ∆, with limits

a = −0.5 ◦C, b = 0.5 ◦C.

NOTE There is no information about the reliability of the limits and so a U-shaped distribution with exactly known limits isassigned. As in 9.5.2.6.2 note, such information may have been omitted from the description in the GUM.

9.5.2.9 Difference δα in expansion coefficients


The estimated bounds on the variability of δα are ±1× 10−6 ◦C−1, with an equal probability of δα havingany value within those bounds [GUM:1995 H.1.3.5]. These bounds are deemed to be reliable to 10 %, givingν(u(δα)) = 50 [GUM:1995 H.1.6].


Assign a rectangular distribution with inexactly prescribed limits (see 6.4.3) to δα, with

a = −1.0× 10−6 ◦C−1, b = 1.0× 10−6 ◦C−1, d = 0.1× 10−6 ◦C−1.

The stated reliability of 10 % on the estimated bounds provides the basis for this value of d.

9.5.2.10 Difference δθ in temperatures


The reference standard and the gauge block being calibrated are expected to be at the same temperature, butthe temperature difference δθ could lie with equal probability anywhere in the estimated interval −0.05 ◦Cto 0.05 ◦C [GUM:1995 H.1.3.6]. This difference is believed to be reliable only to 50 %, giving ν(u(δθ)) = 2[GUM:1995 H.1.6].


Assign a rectangular distribution with inexactly prescribed limits (see 6.4.3) to δθ, with

a = −0.050 ◦C, b = 0.050 ◦C, d = 0.025 ◦C.

The stated reliability of 50 % provides the basis for this value of d.

9.5.3 Propagation and summarizing

9.5.3.1 The GUM uncertainty framework

The application of the GUM uncertainty framework is based on

— a first-order Taylor series approximation to the model (36) or (37),

— use of the Welch-Satterthwaite formula to evaluate an effective degrees of freedom (rounded towards zero) asso-ciated with the uncertainty obtained from the law of propagation of uncertainty, and

— assigning a scaled and shifted t-distribution with the above degrees of freedom to the output quantity.


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9.5.3.2 Monte Carlo method

The application of MCM

— requires sampling from a rectangular distribution (see 6.4.2.4 and C.3.3), Gaussian distribution (see 6.4.7.4and C.4), t-distribution (see 6.4.9.5 and C.6), U-shaped distribution (see 6.4.6.4), and rectangular distributionwith inexactly prescribed limits (see 6.4.3.4), and

— implements adaptive MCM (see 7.9) with a numerical tolerance (δ = 0.5) set to deliver ndig = 2 significantdecimal digits in the standard uncertainty.

9.5.4 Results

9.5.4.1 Table 11 gives the results obtained for the approximate model (37) using the information summarized intable 10. Figure 17 shows the PDFs for δL obtained from the application of the GUM uncertainty framework (solidcurve) and MCM (scaled frequency distribution). The distribution obtained from the GUM uncertainty framework isa t-distribution with ν = 16 degrees of freedom. The endpoints of the shortest 99 % coverage intervals for δL obtainedfrom the PDFs, which are indicated as vertical lines, are visually indistinguishable.

9.5.4.2 1.26×106 trials were taken by the adaptive Monte Carlo procedure. The calculations were also carried outfor a coverage probability of 95 %, for which 0.53× 106 trials were taken.

Table 11 — Results obtained for the approximate model (37) using the information summarized in table 10

(9.5.4.1, 9.5.4.3)

Method δL u(δL) Shortest 99 % coverage/nm /nm interval for δL /nm

GUF 838 32 [745, 931]MCM 838 36 [745, 932]

Figure 17 — PDFs for δL obtained using the GUM uncertainty framework (continuous bell-shaped curve)

and MCM (scaled histogram) for the approximate model (37) using the information summarized in table 10

(9.5.4.1)

9.5.4.3 Results obtained for the non-linear model (36) are identical to the results in table 11 to the number ofdecimal digits given there.

9.5.4.4 There are some modest differences in the results obtained. u(δL) was 4 nm greater for the applicationof MCM than for the GUM uncertainty framework. The length of the 99 % coverage interval for δL was 1 nm greater.


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These results apply equally to the non-linear and the approximate models. Whether such differences are importanthas to be judged in terms of the manner in which the results are to be used.


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Annex AHistorical perspective

A.1 The GUM is a rich document, covering many aspects of uncertainty evaluation. Although it does notrefer explicitly to the use of a Monte Carlo method, such use was recognized during the drafting of the GUM.The ISO/IEC/OIML/BIPM draft (First Edition) of June 1992, produced by ISO/TAG 4/WG 3, states [G.1.5]:

If the relationship between Y and its input quantities is nonlinear, or if the values available for the parameterscharacterizing the probabilities of the Xi (expectation, variance, higher moments) are only estimates and arethemselves characterized by probability distributions, and a first-order Taylor expansion of the relationshipis not an acceptable approximation, the distribution of Y cannot be expressed as a convolution. In this case,a numerical approach (such as Monte Carlo calculations) will generally be required and the evaluation iscomputationally more difficult.

A.2 In the published version of the GUM, this subclause had been modified to read:

If the functional relationship between Y and its input quantities is nonlinear and a first-order Taylor expansionof the relationship is not an acceptable approximation (see 5.1.2 and 5.1.5), then the probability distributionof Y cannot be obtained by convolving the distributions of the input quantities. In such cases, other analyticalor numerical methods are required.

A.3 The interpretation made here of this re-wording is that “other analytical or numerical methods” cover anyother appropriate approach. This interpretation is consistent with that of the National Institute of Standards andTechnology of the United States [50]:

[6.6] The NIST policy provides for exceptions as follows (see Annex C):

It is understood that any valid statistical method that is technically justified under the existing circumstancesmay be used to determine the equivalent of ui, uc, or U . Further, it is recognized that international, national,or contractual agreements to which NIST is a party may occasionally require deviation from NIST policy. Inboth cases, the report of uncertainty must document what was done and why.


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Annex BSensitivity coefficients and uncertainty budgets

B.1 Neither the propagation of distributions nor its implementation using MCM provides sensitivity coeffi-cients [GUM:1995 5.1.3]. However, by holding all input quantities but one fixed at their best estimates, MCMcan be used to provide the PDF for the output quantity for the model having just that input quantity as a vari-able [8]. The ratio of the standard deviation of the resulting model values (cf. 7.6) and the standard uncertaintyassociated with the best estimate of the relevant input quantity can be taken as a sensitivity coefficient. This ratiocorresponds to that which would be obtained by taking all higher-order terms in the Taylor series expansion of themodel into account. This approach may be viewed as a generalization of the approximate partial-derivative formulain the GUM [GUM:1995 5.1.3 note 2]. Both the sensitivity coefficients and the contributions for each input quantityto the uncertainty associated with the estimate of the output quantity will in general differ from those obtained withthe GUM.

B.2 In many measurement contexts it is common practice to list the uncertainty components ui(y) = |ci|u(xi),i = 1, . . . , N , where ci is the ith sensitivity coefficient and u(xi) the standard uncertainty associated with the ith inputestimate xi, contributing to the standard uncertainty u(y). Usually these are presented in a table, the “uncertaintybudget”. This practice may be useful to identify the dominant terms contributing to u(y) associated with the estimateof the output quantity. However, in cases for which (a valid implementation of) the propagation of distributions ismore appropriate, an uncertainty budget should be regarded as a qualitative tool.


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Annex CSampling from probability distributions

C.1 General

C.1.1 This annex provides technical information concerning sampling from probability distributions. Such samplingforms a central part of the use of MCM as an implementation of the propagation of distributions. A digital library ofmathematical functions [38] and a repository of relevant software [37] may also be consulted.

C.1.2 A generator for any distribution, such as the distributions considered in 6.4 (also see table 1), can in principlebe obtained from its distribution function, together with the use of a generator for the rectangular distribution, asindicated in C.2. A generator for a rectangular distribution is provided in C.3.3. For some distributions, such as theGaussian distribution and the t-distribution, it is more efficient to use specifically developed generators, such as thoseprovided in this annex. Subclause 6.4 also gives advice on sampling from probability distributions.

NOTE Generators other than those given in this annex can be used. Their statistical quality should be tested before use. Atesting facility is available for pseudo-random number generators for the rectangular distribution. See C.3.2.

C.2 General distributions

A draw from any strictly increasing, univariate continuous distribution function GX(ξ) can be made by transforminga draw from a rectangular distribution:

a) draw a random number ρ from the rectangular distribution R(0, 1);

b) determine ξ satisfying GX(ξ) = ρ.

NOTE 1 The inversion required in step b), that is forming ξ = G−1X (ρ), may be possible analytically. Otherwise it can be

carried out numerically.

EXAMPLE As an instance of analytical inversion, consider the exponential PDF for X, with X having expectation x (> 0),viz. gX(ξ) = exp(−ξ/x)/x, for ξ ≥ 0, and zero otherwise (see 6.4.10). Then, by integration, GX(ξ) = 1− exp(−ξ/x), for ξ ≥ 0,and zero otherwise. Hence ξ = −x ln(1− ρ). This result can be simplified slightly by using the fact that if a variable Q has therectangular distribution R(0, 1), then so has 1−Q. Hence, ξ = −x ln ρ.

NOTE 2 Numerically, ξ can generally be determined by solving the “zero-of-a-function” problem GX(ξ)− ρ = 0. Upper andlower bounds for ξ are typically easily found, in which case a recognized “bracketing” algorithm such as bisection or, moreefficiently, a combination of linear interpolation and bisection [11], for example, can be used to determine ξ.

NOTE 3 If the pseudo-random number generator for the rectangular distribution is to be used as a basis for generatingnumbers from another distribution, a draw of ρ equal to zero or one can cause failure of that generator. An example is theexponential distribution (see 6.4.10). Its PDF (expression (9)) is not defined for ρ equal to zero or one. The use of the generatorgiven in C.3.3 would not give rise to a failure of that type.

C.3 Rectangular distribution

C.3.1 General

C.3.1.1 The ability to generate pseudo-random numbers from a rectangular distribution is fundamental in its ownright, and also as the basis for generating pseudo-random numbers from any distribution (see C.2, C.4 and C.6) usingan appropriate algorithm or formula. In the latter regard, the quality of the numbers generated from a non-rectangulardistribution depends on that of the generator of numbers from a rectangular distribution and on the properties ofthe algorithm employed. The quality of the numbers generated from a non-rectangular distribution can therefore beexpected to be related to those generated from the rectangular distribution. Only a generator that can faithfullyprovide rectangularly distributed numbers used in conjunction with a good algorithm can be expected to constitute agenerator that can faithfully provide non-rectangularly distributed numbers.


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C.3.1.2 It is thus important that the underlying facility for generating rectangularly distributed numbers issound [31]. Unless the user is sure of its pedigree, a generator should not be used until adequate testing has beencarried out. Misleading results can otherwise be obtained. The use of a testing facility [30] is recommended. Aprocedure for generating rectangularly distributed numbers, which has been shown to perform well in these tests andis straightforward to implement, is given in C.3.3.

C.3.1.3 Table C.1 defines relevant aspects of the functioning of a procedure for generating pseudo-random numbersfrom the rectangular distribution R(0, 1), specifying the input, input-output and output parameters associated withtheir determination.

NOTE 1 By setting the seeds in table C.1 to seeds previously used, the same sequence of random numbers can be produced.Doing so is important as part of software regression testing, used to verify the consistency of results produced using the softwarewith those from previous versions.

NOTE 2 Some pseudo-random number generators provide a single draw at each call and others several draws.

Table C.1 — Generation of pseudo-random numbers from a rectangular distribution (C.3.1.3, C.3.2.2)

Input parameter

q Number of pseudo-random numbers to be generated

Input-output parameter

t Column vector of parameters, some of which may be required as input quantities, that may be changed as part of thecomputation. Subsequent values of these parameters are not usually of immediate concern to the user. The parametersare needed to help control the process by which the pseudo-random numbers are generated. The parameters maybe realized as global variables and thus not explicitly appear as parameters of the procedure. One or more of theseparameters may be a seed, used to initiate the sequence of random numbers produced by successive calls of the procedure

Output parameter

z Column vector of q draws from the rectangular distribution R(0, 1)

C.3.1.4 A pseudo-random number x drawn from R(a, b) is given by a + (b − a)z, where z is a pseudo-randomnumber drawn from R(0, 1).

C.3.2 Randomness tests

C.3.2.1 Any pseudo-random number generator used should

a) have good statistical properties,

b) readily be implemented in any programming language, and

c) give the same results for the same seed on any computer.

It is also desirable that it is compact, thus rendering its implementation straightforward. One such generator thatcomes close to satisfying these requirements is that due to Wichmann and Hill [52, 53]. It has been used in many areasincluding uncertainty computations. However, its cycle length (the number of random numbers generated before thesequence is repeated) is 231, today considered inadequate for some problems. Moreover, not all tests of its statisticalproperties were passed [35]. Further, the generator was designed for 16-bit computers, whereas today 32-bit and 64-bitcomputers are almost universally used.

NOTE The period of the sequence of numbers produced by a pseudo-random number generator is the number of consecutivenumbers in the sequence before they are repeated.

C.3.2.2 An extensive test of the statistical properties of any generator submitted to it is carried out by thetest suite TestU01 [30]. This suite is very detailed, with many individual tests, including the so-called Big Crush.Several generators passing the Big Crush test are listed by Wichmann and Hill [54]. An enhanced Wichmann-Hillgenerator (see C.3.3) also passes the test, and has the properties [54] that


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a) it is straightforward to code in any programming language. It does not depend upon bit manipulation used bysome generators,

b) the state (the amount of information preserved by the generator between calls to it) is small and easy to handle(cf. the parameter t in table C.1),

c) it can readily be used to provide multiple sequences needed for highly parallel applications, likely to be a featureof future uncertainty calculations, and

d) there are variants of the generator for 32- and 64-bit computers.

C.3.3 Procedure for generating pseudo-random numbers from a rectangular distribution

C.3.3.1 Like the previous generator, the enhanced Wichmann-Hill generator is a combination of congruentialgenerators. The new generator combines four such generators, whereas the previous version combined three. The newgenerator has a period of 2121, acceptable for any conceivable application.

C.3.3.2 Table C.2 defines the enhanced Wichmann-Hill generator for producing pseudo-random numbersfrom R(0, 1) for a 32-bit computer.

Table C.2 — The enhanced Wichmann-Hill generator for pseudo-random numbers (C.3.3.2, C.3.3.3) from a

rectangular distribution on the interval (0, 1) for 32-bit computers. bwc denotes the largest integer no greater

than w. ij mod bj denotes the remainder on division of ij by bj

Input parameter

None

Input-output parameter

i1,i2,i3,i4

Integer parameters required as input quantities and that are changed by the procedure. Set to integers between 1and 2 147 483 647 before the first call. Do not disturb between calls. Subsequent values of these parameters are notusually of concern to the user. The parameters provide the basis by which the pseudo-random numbers are generated.They may be realized as global variables and thus not appear explicitly as parameters of the procedure

Constant

a, Vectors of integer constants of dimension 1× 4, where a = (a1, . . . , a4), etc., given by:

b, a = (11 600, 47 003, 23 000, 33 000),

c b = (185 127, 45 688, 93 368, 65 075),

d, c = (10 379, 10 479 , 19 423, 8 123),

d = 2 147 483 123× (1, 1, 1, 1) + (456, 420, 300, 0). Do not disturb between calls

Output parameter

r Pseudo-random number drawn from R(0, 1)

Computation

a) For j = 1, . . . , 4:

i) Form ij = aj × (ij mod bj)− cj × bij/bjcii) If ij < 0, replace ij by ij + dj

b) Form w =∑4

j=1ij/dj

c) Form r = w − bwc

C.3.3.3 For 64-bit computers, step a) of Computation, including (i) and (ii), in the generator of table C.2 is to bereplaced by the simpler step:

a) For j = 1, . . . , 4, form ij = (aj × ij) mod dj


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C.4 Gaussian distribution

The procedure in table C.3 provides draws from the standard Gaussian distribution N(0, 1) using the Box-Mullertransform [3]. A draw from the Gaussian distribution N(µ, σ2) is given by µ + σz, where z is a draw from N(0, 1).

Table C.3 — The Box-Muller Gaussian pseudo-random number generator (C.4)

Input parameter

None

Output parameter

z1,z2

Two draws, obtained independently, from a standard Gaussian distribution

Computation

a) Generate random draws r1 and r2 independently from the rectangular distribution R(0, 1)

b) Form z1 =√−2 ln r1 cos 2πr2 and z2 =

√−2 ln r1 sin 2πr2

C.5 Multivariate Gaussian distribution

C.5.1 The most important multivariate distribution is the multivariate (or joint) Gaussian distribution N(µ, V ),where µ is a vector of expectations of dimension n× 1 and V a covariance matrix of dimension n× n.

C.5.2 Draws from N(µ, V ) [45, 49] can be obtained using the procedure in table C.4.

NOTE 1 If V is positive definite (i.e. all its eigenvalues are strictly positive), the Cholesky factor R is unique [23, page 204].

NOTE 2 If V is not positive definite, perhaps because of numerical rounding errors or other sources, R may not exist.Moreover, in cases where one or more of the eigenvalues of V is very small (but positive), the software implementation of theCholesky factorization algorithm used may be unable to form R because of the effects of floating-point errors. In either ofthese situations it is recommended that V is “repaired”, i.e. as small a change as possible is made to V such that the Choleskyfactor R for the modified matrix is well defined. The resulting factor is exact for a covariance matrix that numerically is closeto the original V . A simple repair procedure is available [49, page 322] for this purpose, and is embodied in the MULTNORMgenerator [45].

NOTE 3 If V is semi-positive definite, the eigendecomposition V = QΛQ>, where Q is an orthogonal matrix and Λ adiagonal matrix, can be formed. Then Λ1/2Q> can be used to obtain draws from N(0, V ), even if V is rank deficient.

Table C.4 — A multivariate Gaussian random number generator (C.5.2)

Input parameter

n Dimension of the multivariate Gaussian distribution

µ Vector of expectations of dimension n× 1

V Covariance matrix of dimension n× n

q Number of multivariate Gaussian pseudo-random numbers to be generated

Output parameter

X Matrix of dimension n× q, the jth column of which is a draw from the multivariate Gaussian distribution

Computation

a) Form the Cholesky factor R of V , i.e. the upper triangular matrix satisfying V = R>R. (To generate q pseudo-random numbers, it is necessary to perform this matrix factorization only once.)

b) Generate an array Z of standard Gaussian variates of dimension n× q

c) Form

X = µ1> + R>Z,where 1 denotes a vector of ones of dimension q × 1


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C.5.3 Figure C.1 shows 200 points generated using the MULTNORM generator [45] from N(µ, V ), where

µ =

2.0

3.0

, V =

2.0 1.9

1.9 2.0

,

i.e. in which the two quantities concerned are positively correlated. Similar generators are available elsewhere [12].

Figure C.1 — Points sampled from a bivariate Gaussian distribution with positive correlation (C.5.3, C.5.4)

C.5.4 In figure C.1, the points span an elongated angled ellipse. Were the off-diagonal elements of V to be replacedby zero, the points would span a circle. Were the diagonal elements made unequal, and the off-diagonal elements keptat zero, the points would span an ellipse whose axes were parallel to the axes of the graph. If the diagonal elementswere negative, and hence the quantities concerned negatively correlated, the major axis of the ellipse would have anegative rather than a positive gradient.

C.6 t-distribution

The procedure in table C.5 provides an approach [29], [44, page 63] to obtain draws from the t-distribution with νdegrees of freedom.

Table C.5 — A t-distribution pseudo-random number generator (C.6)

Input parameter

ν Degrees of freedom

Output parameter

t Draw from a t-distribution with ν degrees of freedom

Computation

a) Generate two draws r1 and r2 independently from the rectangular distribution R(0, 1)

b) If r1 < 1/2, form t = 1/(4r1 − 1) and v = r2/t2; otherwise form t = 4r1 − 3 and v = r2

c) If v < 1− |t|/2 or v < (1 + t2/ν)−(ν+1)/2, accept t as a draw from the t-distribution; otherwise repeat from step a)

NOTE ν must be greater than two for the standard deviation of the t-distribution with ν degrees of freedom to be finite.


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Annex DContinuous approximation to the distribution function for the output quantity

D.1 It is sometimes useful to work with a continuous approximation GY (η), say, to the distribution function forthe output quantity Y , rather than the discrete representation G of 7.5.

NOTE Working with a continuous approximation means, for instance, that

a) sampling from the distribution function can be carried out without the need for rounding, as in the discrete case, and

b) numerical methods that require continuity for their operation can be used to determine the shortest coverage interval.

D.2 In order to form GY (η), consider the discrete representation G = {y(r), r = 1, . . . ,M} of GY (η) in 7.5.1, afterreplacing replicate model values of y(r) as necessary (step b) in that subclause). Then, carry out the following steps:

a) assign uniformly spaced cumulative probabilities pr = (r − 1/2)/M , r = 1, . . . ,M , to the y(r) [8]. The numericalvalues pr, r = 1, . . . ,M , are the midpoints of M contiguous probability intervals of width 1/M between zero andone;

b) form GY (η) as the (continuous) strictly increasing piecewise-linear function joining the M points (y(r), pr),r = 1, . . . ,M :

GY (η) =r − 1/2

M+

η − y(r)

M(y(r+1) − y(r)), y(r) ≤ η ≤ y(r+1), r = 1, . . . ,M − 1. (D.1)

NOTE The form (D.1) provides a convenient basis for sampling from GY (η) for purposes of a further stage of uncertaintyevaluation. See C.2 for sampling inversely from a distribution function. Some software libraries and packages provide facilities

for piecewise-linear interpolation. Since GY (η) is piecewise linear, so is its inverse, and such facilities can readily be applied.

D.3 Figure D.1 illustrates GY (η) obtained using MCM based on M = 50 sampled values from a Gaussian PDF gY (η)with Y having expectation 3 and standard deviation 1.

Figure D.1 — An approximation GY (η) to the distribution function GY (η) (D.3). “Unit” denotes any unit

D.4 Consider gY (η) = G′Y (η), with GY (η) given in expression (D.1). The function gY (η) is piecewise constant with

breakpoints at η = y(1), . . . , y(M). The expectation y and standard deviation u(y) of Y , described by gY (η), are taken,respectively, as an estimate of Y and the standard uncertainty associated with that estimate. y and u(y) are given by

y =1M

M∑r=1

′′ y(r) (D.2)


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and

u2(y) =1M

(M∑

r=1

′′ (y(r) − y)2 − 16

M−1∑r=1

(y(r+1) − y(r))2)

, (D.3)

where the double prime on a summation symbol indicates that the first and the last terms in the sum are to be takenwith weight one half.

NOTE For a sufficiently large numerical value of M (105, say, or greater), y and u(y) obtained using formulæ (D.2) and (D.3)would generally be indistinguishable for practical purposes from those given by formulæ (16) and (17), respectively.

D.5 Let α denote any value between zero and 1− p, where p is the required coverage probability (e.g. 0.95). Theendpoints of a 100p % coverage interval can be obtained from G(η) by inverse linear interpolation. To determine thelower endpoint ylow such that α = GY (ylow), identify the index r for which the points (y(r), pr) and (y(r+1), pr+1)satisfy

pr ≤ α < pr+1.

Then, by inverse linear interpolation,

ylow = y(r) +(y(r+1) − y(r)

) α− pr

pr+1 − pr.

Similarly, the upper endpoint yhigh, determined such that p + α = GY (yhigh), is calculated from

yhigh = y(s) +(y(s+1) − y(s)

) p + α− ps

ps+1 − ps,

where the index s is such that the points (y(s), ps) and (y(s+1), ps+1) satisfy

ps ≤ p + α < ps+1.

D.6 The choice α = 0.025 gives the coverage interval defined by the 0.025- and 0.975-quantiles. This choice providesthe probabilistically symmetric 95 % coverage interval for Y .

D.7 The shortest coverage interval can generally be obtained from GY (η) by determining α such thatG−1

Y (p + α)− G−1Y (α), = H(α), say, is a minimum. A straightforward numerical approach to determining the min-

imum is to evaluate H(α) for a large number of uniformly spaced choices {αk} of α between zero and 1 − p, andchoose α` from the set {αk} that yields the minimum from the set {H(αk)}.

D.8 The computation of a coverage interval is facilitated if pM is an integer. Then, the numerical value of α,such that H(α) is a minimum, is equal to r∗/M , where r∗ is the index r such that the interval length y(r+pM) − y(r),over r = 1, . . . , (1− p)M , is least.


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Annex ECoverage interval for the four-fold convolution of a rectangular distribution

E.1 In 9.2.3.2, the analytic solution

± 2√

3[2− (3/5)1/4] ≈ ±3.88 (E.1)

is stated. It constitutes the endpoints of the probabilistically symmetric 95 % coverage interval for the outputquantity Y in an additive model having four input quantities with expectations of zero and standard deviations ofunity, the PDFs for which are identical rectangular distributions. This result is established in this annex.

E.2 The rectangular distribution R(a, b) (see 6.4.2) takes the constant value (b − a)−1 for a ≤ ρ ≤ b and is zerootherwise. The n-fold convolution of R(0, 1) is the B-spline Bn(ρ) of order n (degree n − 1) with knots 0, . . . , n [46].An explicit expression is [6]

Bn(ρ) =1

(n− 1)!

n∑r=0

nCr(−1)r(ρ− r)n−1+ ,

where

nCr =n!

r!(n− r)!, z+ = max(z, 0).

In particular,

B4(ρ) =16ρ3, 0 ≤ ρ ≤ 1

(with different cubic polynomial expressions for B4(ρ) in other intervals between adjacent knots), and hence

∫ 1

0

B4(ρ) dρ =[

124

ρ4

]10

=124

≈ 0.0417.

E.3 The left-hand endpoint ylow of the probabilistically symmetric 95 % coverage interval lies between zero andone, since

0.025 =140

<124

of the area under the PDF lies to the left of ylow, which is therefore given by∫ ylow

0

B4(ρ) dρ =124

y4low =

140

,

i.e.

ylow = (3/5)1/4.

By symmetry, the right-hand endpoint is

yhigh = 4− (3/5)1/4.

Thus, the probabilistically symmetric 95 % coverage interval is[(3/5)1/4, 4− (3/5)1/4

]≡ 2±

(2− (3/5)1/4

).


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The corresponding coverage interval for the four-fold convolution of the rectangular PDF R(−√

3,√

3) (which has zeroexpectation and unit standard deviation) is given by shifting this result by two units and scaling it by 2

√3 units,

yielding expression (E.1).


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Annex FComparison loss problem

This annex is concerned with some details of the comparison loss problem (see 9.4). Subclause F.1 provides theexpectation and standard deviation of δY (see 9.4.2.1.2). Subclause F.2 provides the PDF for δY analyticallywhen x1 = x2 = r(x1, x2) = 0 (see 9.4.2.1.2). Subclause F.3 applies the GUM uncertainty framework for uncor-related and correlated input quantities (see 9.4.2.1.3 and 9.4.3.1.1).

F.1 Expectation and standard deviation obtained analytically

F.1.1 The variance of a quantity X can be expressed in terms of expectations as [42, page 124]

V (X) = E(X2)− [E(X)]2.

Thus,

E(X2) = [E(X)]2 + V (X) = x2 + u2(x),

where x is the best estimate of X and u(x) the standard uncertainty associated with x. Thus, for the model (28),viz. δY = 1− Y = X2

1 + X22 ,

δy = E(δY ) = x21 + x2

2 + u2(x1) + u2(x2).

This result applies

a) regardless of which PDFs are assigned to X1 and X2, and

b) whether X1 and X2 are independent or not.

F.1.2 The standard uncertainty associated with δy can be obtained from

u2(δy) = u2(x21) + u2(x2

2) + 2u(x21, x

22),

where, for i = 1 and i = 2, u2(x2i ) = V (X2

i ), and u(x21, x

22) = Cov(X2

1 , X22 ). Then, applying Price’s Theorem for

Gaussian distributions [40, 41],

u2(δy) = 4u2(x1)x21 + 4u2(x2)x2

2 + 2u4(x1) + 2u4(x2) + 4u2(x1, x2) + 8u(x1, x2)x1x2. (F.1)

When x2 = 0 and u(x2) = u(x1), and replacing u(x1, x2) by r(x1, x2)u2(x1),

u(δy) = 2{x2

1 + [1 + r2(x1, x2)]u2(x1)}1/2

u(x1).

F.1.3 When X1 and X2 are uncorrelated, i.e. u(x1, x2) = 0, expression (F.1) becomes

u2(δy) = 4u2(x1)x21 + 4u2(x2)x2

2 + 2u4(x1) + 2u4(x2). (F.2)

Expression (F.2) can be verified by applying formula (10) of the GUM [GUM:1995 5.1.2] and the immediately follow-ing GUM formula [GUM:1995 5.1.2 note].


JCGM 101:2008

F.2 Analytic solution for zero estimate of the voltage reflection coefficient having associatedzero covariance

F.2.1 For the case x1 = x2 = r(x1, x2) = 0 and u(x1) = u(x2), the PDF gY (η) for Y can be obtained analytically.It is valuable to have such a solution for further validation purposes. In the above circumstances,

δY = u2(x1)[

X21

u2(x1)+

X22

u2(x2)

].

F.2.2 The term in square brackets is the sum, Z, say, of the squares of two independent quantities, each of whichis distributed as a standard Gaussian PDF. Thus the sum is distributed as chi-squared with two degrees of freedom[42, page 177], so that

δY = u2(x1)Z,

where Z has PDF

gZ(z) = χ22(z) = e−z/2/2.

F.2.3 The application of a general formula [42, pages 57–61] for the PDF gY (η) of a differentiable and strictlydecreasing function of a variable (here Z) with a specified PDF yields

gY (η) =1

u2(x1)χ2

2

(η

u2(x1)

)=

12u2(x1)

exp(− η

2u2(x1)

), η ≥ 0.

F.2.4 The expectation of δY is given by

δy = E(δY ) =∫ ∞

0

ηgY (η) dη = 2u2(x1)

and the variance

u2(δy) = V (δY ) =∫ ∞

0

(η − y)2gY (η) dη = 4u4(x1),

i.e. the standard deviation is 2u2(x1), results that are consistent with those in F.1.

F.2.5 By integration, the corresponding distribution function is

GY (η) = 1− exp(− η

2u2(x1)

), η ≥ 0. (F.3)

F.2.6 Let δyα be that η in expression (F.3) corresponding to GY (η) = α for any α satisfying 0 ≤ α ≤ 1− p. Then

δyα = −2u2(x1) ln(1− α)

and a 100p % coverage interval for δY (see 7.7) is

[δyα, δyp+α] ≡ [−2u2(x1) ln(1− α), − 2u2(x1) ln(1− p− α)] (F.4)

with length

H(α) = −2u2(x1) ln(

1− p

1− α

).

F.2.7 The shortest 100p % coverage interval is given by determining α to minimize H(α) (see 5.3.4). Since H(α)is a strictly increasing function of α for 0 ≤ α ≤ 1− p, H(α) is minimized when α = 0. Thus, the shortest 100p %


JCGM 101:2008

coverage interval for δY is

[0, − 2u2(x1) ln(1− p)].

For u(x1) = 0.005, the shortest 95 % coverage interval is

[0, 0.000 149 8].

F.2.8 The 95 % probabilistically symmetric coverage interval for δY is given by setting α = (1− p)/2 (see 5.3.3):

[−2u2(x1) ln 0.975, − 2u2(x1) ln 0.025] = [0.000 001 3, 0.000 184 4],

which is 20 % longer than the shortest 95 % coverage interval.

NOTE The above analysis is indicative of an analytical approach that can be applied to some problems of this type. In thisparticular case, the results could in fact have been obtained more directly, since gY (η) is strictly increasing and the shortestcoverage interval is always in the region of highest density.

F.3 GUM uncertainty framework applied to the comparison loss problem

F.3.1 Uncorrelated input quantities

F.3.1.1 The comparison loss problem considered in 9.4 has as the model of measurement

δY = f(X) = f(X1, X2) = X21 + X2

2 ,

where X1 and X2 are assigned Gaussian PDFs having expectations x1 and x2 and variances u2(x1) and u2(x2),respectively.

F.3.1.2 The application of GUM subclause 5.1.1 gives

δy = x21 + x2

2

as the estimate of δY . The only non-trivially non-zero partial derivatives of the model are, for i = 1, 2,

∂f

∂Xi= 2Xi,

∂2f

∂X2i

= 2.

F.3.1.3 Hence the application of GUM subclause 5.1.2 gives, for the standard uncertainty u(δy),

u2(δy) =

[(∂f

∂X1

)2

u2(x1) +(

∂f

∂X2

)2

u2(x2)

]∣∣∣∣∣X=x

= 4x21u

2(x1) + 4x22u

2(x2), (F.5)

based on a first-order Taylor series approximation of f(X). If the non-linearity of f is significant [GUM:1995 5.1.2 note],the term

12

[∂2f

∂X21

+∂2f

∂X22

]∣∣∣∣X=x

u2(x1)u2(x2)

needs to be appended to formula (F.5), in which case formula (F.5) becomes

u2(δy) = 4x21u

2(x1) + 4x22u

2(x2) + 4u2(x1)u2(x2). (F.6)

F.3.1.4 A 95 % coverage interval for δY is given by

δy ± 2u(δy),

as a consequence of δY having a Gaussian PDF.


JCGM 101:2008

F.3.2 Correlated input quantities

F.3.2.1 When the input quantities are correlated, the uncertainty matrix associated with the best estimates of theinput quantities is given in formulæ (27).

F.3.2.2 The application of GUM subclause 5.2.2 gives

u2(δy) =

[(∂f

∂X1

)2

u2(x1) +(

∂f

∂X2

)2

u2(x2) + 2∂f

∂X1

∂f

∂X2r(x1, x2)u(x1)u(x2)

]∣∣∣∣∣X=x

= 4x21u

2(x1) + 4x22u

2(x2) + 8r(x1, x2)x1x2u(x1)u(x2). (F.7)


JCGM 101:2008

Annex GGlossary of principal symbols

A random variable representing the lower limit of a rectangular distribution with inexactly prescribedlimits

a lower limit of the interval in which a random variable is known to lie

a midpoint of the interval in which the lower limit A of a rectangular distribution with inexactly pre-scribed limits is known to lie

B random variable representing the upper limit of a rectangular distribution with inexactly prescribedlimits

b upper limit of the interval in which a random variable is known to lie

b midpoint of the interval in which the upper limit B of a rectangular distribution with inexactlyprescribed limits is known to lie

CTrap(a, b, d) rectangular distribution with inexactly prescribed limits (curvilinear trapezoid distribution) with pa-rameters a, b, and d

Cov(Xi, Xj) covariance for two random variables Xi and Xj

c ndig-decimal digit integer

ci ith sensitivity coefficient, obtained as the partial derivative of the model f for the measurement withrespect to the ith input quantity Xi evaluated at the vector estimate x of the vector input quantity X

d semi-width of the intervals in which the lower and upper limits A and B of a rectangular distributionwith inexactly prescribed limits are known to lie

dhigh absolute value of the difference between the right-hand endpoints of the coverage intervals providedby the GUM uncertainty framework and a Monte Carlo method

dlow absolute value of the difference between the left-hand endpoints of the coverage intervals provided bythe GUM uncertainty framework and a Monte Carlo method

e base of the natural logarithm

E(X) expectation of a random variable X

E(X) vector expectation of a vector random variable X

E(Xr) rth moment of a random variable X

Ex(λ) exponential distribution with parameter λ

f mathematical model of measurement, expressed as a functional relationship between an output quan-tity Y and the input quantities X1, . . . , XN on which Y depends

G discrete representation of the distribution function GY (η) for the output quantity Y from a MonteCarlo procedure

G(α, β) gamma distribution with parameters α and β


JCGM 101:2008

gX(ξ) probability density function with variable ξ for the input quantity X

gX(ξ) joint (multivariate) probability density function with vector variable ξ for the vector input quantity X

gXi(ξi) probability density function with variable ξi for the input quantity Xi

GY (η) distribution function with variable η for the output quantity Y

GY (η) continuous approximation to the distribution function GY (η) for the output quantity Y

gY (η) probability density function with variable η for the output quantity Y

gY (η) derivative of GY (η) with respect to η, providing a numerical approximation to the probability densityfunction gY (η) for the output quantity Y

J smallest integer greater than or equal to 100/(1− p)

kp coverage factor corresponding to the coverage probability p

` integer in the representation c× 10` of a numerical value, where c is an ndig-decimal digit integer

M number of Monte Carlo trials

N number of input quantities X1, . . . , XN

N(0, 1) standard Gaussian distribution

N(µ, σ2) Gaussian distribution with parameters µ and σ2

N(µ,V ) multivariate Gaussian distribution with parameters µ and V

n number of indications in a series

ndig number of significant decimal digits regarded as meaningful in a numerical value

Pr(z) probability of the event z

p coverage probability

q integer part of pM + 1/2

q number of objects counted in a sample of specified size

R upper triangular matrix

R(0, 1) standard rectangular distribution over the interval [0, 1]

R(a, b) rectangular distribution over the interval [a, b]

r(xi, xj) correlation coefficient associated with the estimates xi and xj of the input quantities Xi and Xj

s standard deviation of a series of n indications x1, . . . , xn

sp pooled standard deviation obtained from several series of indications

> superscript denoting matrix transpose


JCGM 101:2008

sz standard deviation associated with the average z of the values z(1), . . . , z(h) in an adaptive Monte Carloprocedure, where z may denote an estimate y of the output quantity Y , the standard uncertainty u(y)associated with y, or the left-hand endpoint ylow or right-hand endpoint yhigh of a coverage intervalfor Y

T(a, b) triangular distribution over the interval [a, b]

Trap(a, b, β) trapezoidal distribution over the interval [a, b] with parameter β

tν central t-distribution with ν degrees of freedom

tν(µ, σ2) scaled and shifted t-distribution with parameters µ and σ2, and ν degrees of freedom

U(0, 1) standard arc sine (U-shaped) distribution over the interval [0, 1]

U(a, b) arc sine (U-shaped) distribution over the interval [a, b]

Up expanded uncertainty corresponding to a coverage probability p

Ux uncertainty matrix associated with the vector estimate x of the vector input quantity X

u(x) vector (u(x1), . . . , u(xN ))> of standard uncertainties associated with the vector estimate x of thevector input quantity X

u(xi) standard uncertainty associated with the estimate xi of the input quantity Xi

u(xi, xj) covariance associated with the estimates xi and xj of the input quantities Xi and Xj

u(y) standard uncertainty associated with the estimate y of the output quantity Y

u(y) standard uncertainty associated with y

uc(y) combined standard uncertainty associated with the estimate y of the output quantity Y

ui(y) ith uncertainty component of the standard uncertainty u(y) associated with the estimate y of theoutput quantity Y

V covariance (variance-covariance) matrix

V (X) variance of a random variable X

V (X) covariance matrix for the vector random variable X

w semi-width (b− a)/2 of an interval [a, b]

X input quantity, regarded as a random variable

X vector (X1, . . . , XN )> of input quantities, regarded as random variables, on which the output quan-tity Y depends

Xi ith input quantity, regarded as a random variable, on which the output quantity Y depends

x estimate (expectation) of X

x vector estimate (vector expectation) (x1, . . . , xN )> of X

x average of a series of n indications x1, . . . , xn


JCGM 101:2008

xi estimate (expectation) of Xi

xi ith indication in a series

xi,r rth Monte Carlo draw from the probability density function for Xi

xr rth Monte Carlo draw, containing values x1,r, . . . , xN,r, drawn from the probability density functionsfor the N input quantities X1, . . . , XN or from the joint probability density function for X

Y (scalar) output quantity, regarded as a random variable

y estimate (expectation) of Y

y estimate of Y , obtained as the average of the M model values yr from a Monte Carlo run or as theexpectation of Y characterized by the probability density function gY (η)

yhigh right-hand endpoint of a coverage interval for Y

ylow left-hand endpoint of a coverage interval for Y

yr rth model value f(xr)

y(r) rth model value after sorting the yr into increasing order

z(h) hth value in an adaptive Monte Carlo procedure, where z may denote an estimate y of the outputquantity Y , the standard uncertainty u(y) associated with y, or the left-hand endpoint ylow or right-hand endpoint yhigh of a coverage interval for Y

α probability value

α parameter of a gamma distribution

β parameter of a trapezoidal distribution equal to the ratio of the semi-width of the top of the trapezoidto that of the base

β parameter of a gamma distribution

Γ(z) gamma function with variable z

δ numerical tolerance associated with a numerical value

δ(z) Dirac delta function with variable z

η variable describing the possible values of the output quantity Y

λ1 top semi-width of the trapezoid for a trapezoidal distribution

λ2 base semi-width of the trapezoid for a trapezoidal distribution

µ expectation of a quantity characterized by a probability distribution

ν degrees of freedom of a t-distribution or a chi-squared distribution

νeff effective degrees of freedom associated with the standard uncertainty u(y)

νp degrees of freedom associated with a pooled standard deviation sp obtained from several series ofindications


JCGM 101:2008

ξ variable describing the possible values of the random variable X

ξ vector variable (ξ1, . . . , ξN )> describing the possible values of the vector input quantity X

ξi variable describing the possible values of the input quantity Xi

σ standard deviation of a quantity characterized by a probability distribution

σ2 variance (squared standard deviation) of a quantity characterized by a probability distribution

Φ phase of a quantity that cycles sinusoidally

χ2ν chi-squared distribution with ν degrees of freedom


JCGM 101:2008

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Alphabetical indexA

arc sine distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6assignment to a quantity . . . . . . . . 6.4.6.1, 9.5.2.8.2expectation and variance . . . . . . . . . . . . . . . . . . 6.4.6.3probability density function . . . . . . . . . . . . . . . 6.4.6.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.6.4

averageof a set of indications . . . . . . 5.11.2, 6.4.9.2, 6.4.9.4,

9.5.1.3, 9.5.2.3.1of sampled model values . . . . . . . . . . . . . . . . . . . . . . 7.6

BBayes’ theorem . . . . . . . . . . . . . 6.1.1, 6.2, 6.4.9.2, 6.4.11.1Bayesian interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12best estimate

from a calibration certificate . . . . . . . . . . . . . . 6.4.9.7of a non-negative input quantity . . . . . . . . . 6.4.10.1of a vector input quantity . . . . . . . . . . . . . . . . . 6.4.8.1of an input quantity. .5.6.2, 5.11.2, 5.11.5, 6.4.7.1,

6.4.9.4, 9.2.2.1, 9.2.3.1, 9.3.1.4, 9.4.1.3

Ccalibration

certificate . . . . . 6.4.3.4, 6.4.9.7, 9.5.2.2.1, 9.5.2.4.1,9.5.2.5.1

gauge block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3microwave power meter . . . . . . . . . . . . . . . . . . . . . . . 9.4

Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1central limit theorem . . . . . . . . . . . . . 5.7.2, 5.11.6, 9.2.4.5chi-squared distribution . . . . . . 6.4.11.4, 9.4.2.2.3, F.2.2Cholesky decomposition . . . . . . . . . . . . . . . . . 6.4.8.4, C.5.2combined standard uncertainty . . . . . . . . . . . . . . . . . . . 4.10convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4.2, A.1, Ecorrelated input quantities . . . .5.6.3, 9.4.3, C.5.3, F.3.2correlation coefficient . . . . . . . . . . . . . . . . 9.4.1.3, 9.4.2.3.1covariance . . . . . . . . . . . . . . 3.10, 3.11, 5.6.3, 9.4.1.3, 9.4.3covariance matrix . . . .3.11, 4.4, 6.4.8.3, 9.4.1.4, 9.4.1.8,

9.4.3.1.3, C.5coverage factor . . . 5.3.3, 5.11.6, 6.4.9.7, 9.2.2.3, 9.2.4.4,

9.5.2.2.1coverage interval . .3.12, 4.11, 5.1.1, 5.3.2, 5.5.1, 5.11.4,

5.11.6, 8.1.2, 8.1.3, D.5, E, F.2.6, F.3.1.4from GUM uncertainty framework. . . .5.6.2, 5.6.3,

5.7.2, 5.8.2from Monte Carlo method . . . . 5.9.6, 7.2.1, 7.6, 7.7length of . . . . . . . . . . . . . . . . . . 3.14, 9.2.2.8, 9.4.2.2.11probabilistically symmetric. . . . . . . . . . . . . . . . . .3.15,

5.3.3, 5.3.5, 5.3.6, 5.5.1, 7.9.4, 8.1.2, 9.2.2.3,9.2.2.4, 9.2.2.6, 9.2.2.9, 9.2.3.2, 9.2.3.4,9.2.4.4, 9.2.4.5, 9.4.2.2.11, D.6, E.1, F.2.8

shortest . . . . . 3.16, 5.3.4–5.3.6, 5.5.1, 8.1.2, 9.2.2.9,9.3.2.1, 9.3.2.3, 9.4.2.1.2, 9.4.2.2.9, 9.4.2.2.11,9.4.2.3.3, 9.4.3.2.2, 9.5.4.1, D.7, D.8, F.2.7,F.2.8

statistical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.4coverage probability3.13, 4.11, 5.1.1, 5.3.2, 5.5.1, 5.6.3,

5.9.6, 7.2.1, 7.7.2, 7.9.3, 8.1.2, 8.1.3, D.5credible interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12curvilinear trapezoid distribution. . . . . .see rectangular

distribution with inexactly prescribed limits

Ddegree of belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2degrees of freedom . . . . . . 3.5, 5.6.2, 5.6.3, 5.7.2, 5.11.5,

6.4.3.3, 6.4.9.2, 6.4.9.4, 6.4.9.5, 7.6, 9.5.2.4.2,9.5.2.5.1, 9.5.4.1, C.6, F.2.2

effective . . . . . . . 5.6.3, 5.7.2, 5.11.6, 6.4.9.4, 6.4.9.7,9.5.2.2.1, 9.5.3.1

of a pooled standard deviation . . . . . . . . . . . . 6.4.9.6distribution function. . . . . . . . . . . . . . . .3.2, 4.2, 5.3.6, 6.5

for output quantity . . . . . .D, 5.2, 5.9.1, 5.11.4, 7.5,9.3.2.3, 9.4.2.2.9

EErlang distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.11.4Estimate of output quantity

from Monte Carlo method . . . . . . . . . . . . . . . . . . . . 7.6estimate of output quantity. .5.1.1, 5.3.1, 5.5.1, 5.11.4,

5.11.6, 9.4.2.2.7, B.2from GUM uncertainty framework . . . . 4.10, 5.6.2,

5.6.3, 9.4.2.1.3, 9.4.2.2.8, F.3.1.2from Monte Carlo method . . 5.9.6, 7.9.4, 9.4.2.2.8,

D.4expanded uncertainty . . . . 5.3.3, 5.6.3, 6.4.9.7, 9.5.2.2.1expectation. . . . . . . .3.6, 4.1, 4.3, 4.8, 5.1.1, 5.3.1, 5.6.2,

5.6.3, 5.9.6, 6.4.1, 7.5.1, 7.6, 9.2.3.3, 9.3.1.4,9.4.1.4, 9.4.1.8, 9.4.2.1.2, 9.4.2.2.7, 9.4.2.2.11,9.4.2.3.2, 9.4.3.1.3, C.2, D.3, D.4, E.3, F.1.1,F.2.4, F.3.1.1

exponential distribution . . . . . . . . . . 6.4.10, 6.4.11.4, C.2assignment to a quantity . . . . . . . . . . . . . . . . . 6.4.10.1expectation and variance . . . . . . . . . . . . . . . . . 6.4.10.3probability density function . . . . . . . . . . . . . . 6.4.10.2sampling from . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.10.4

Ffinite difference approximation . . . . . . . . . . . . . . . . . . . 5.8.1formulation . . . . . . . see stages of uncertainty evaluation

Ggamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.11

assignment to a quantity . . . . . . . . . . . . . . . . . 6.4.11.1expectation and variance . . . . . . . . . . . . . . . . . 6.4.11.3probability density function . . . . . . . . . . . . . . 6.4.11.2sampling from . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.11.4

gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5, 6.4.9.3Gaussian distribution . 3.4, 5.6–5.8, 5.11.6, 6.4.7, 9.2.2,

9.4.1.8, 9.4.2.2.5, 9.4.2.3.2, D.3

80

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assignment to a quantity 6.4.7.1, 9.3.1.4, 9.5.2.7.2,F.3.1.1

expectation and variance . . . . . . . . . . . . . . . . . . 6.4.7.3probability density function . . . . . . . . . . . 3.4, 6.4.7.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . 6.4.7.4, C.4

GUF. . . . . . . . .see GUM uncertainty framework (GUF)Guide to the expression of uncertainty in measurement

(GUM) . . . . . . . . . . . . . . . . . . . 2, 3, 4.15, 5.6.1, AGUM . . . see Guide to the expression of uncertainty in

measurement (GUM)GUM uncertainty framework (GUF) . . . . . 1, 3.18, 4.15,

5.1.2, 5.4.2, 5.6, 5.10.2, F.3comparison with a Monte Carlo method . . . . . 5.11conditions for linear models . . . . . . . . . . . . . . . . . . . 5.7conditions for non-linear models . . . . . . . . . . . . . . 5.8example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1–9.5validation of . . . . 8.1, 9.1.2, 9.2.2.7, 9.2.3.4, 9.2.4.5,

9.3.2.6with higher-order terms 5.8.1, 8.1.2, 9.1.1, 9.3.2.1,

9.3.2.6, 9.4.2.1.3, 9.4.2.2.1

Iindications, analysis of a series of . . 5.11.2, 6.1.3, 6.2.1,

6.2.2, 6.3.1, 6.4.9.2, 6.4.9.4, 6.4.9.6, 9.5.1.3,9.5.2.3.1, 9.5.2.4.1

input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1, 5.1.1International vocabulary of basic and general terms in

metrology (VIM) . . . . . . . . . . . . . . . . . . 2, 3, 4.15

Kknowledge

about a quantity, incomplete . . . . . . . . . . . . . . . . 6.3.2about a quantity, prior . . . . . . . . . . . . . . . . . . . . .5.11.2about an input quantity . . . . . . . . . . . . . . 5.1.1, 6.1.6

Llevel of confidence . . . . . . . . . . . . . . . . . . . . 3.13, 4.11, 7.9.4

MMCM . . . . . . . . . . . . . . . see Monte Carlo method (MCM)mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4, 5.10.1, 7.5.1model of measurement . . . . . . . . . . . . . . . . . . . . 1, 4.1, 5.1.1

additive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.2.1calibration of gauge block . . . . . . . . . . . . . . . . . 9.5.1.5calibration of mass standards. . . . . . . . . . . . . .9.3.1.3calibration of microwave power meter . . . . . 9.4.1.5linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.2, 5.7non-linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4, 5.8

moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9Monte Carlo method (MCM)3.19, 4.15, 5.2, 5.4.1, 5.9,

6.5, 7, 8.1.1, A.1, B.1, C.1.1, D.3adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9comparison with GUM uncertainty framework5.11computation time. . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.8conditions for application . . . . . . . . . . . . . . . . . . . . 5.10example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1–9.5number of trials . . . . . . . . . . . . . . . . . . . . . . . . 7.2, 7.9.1sampling from probability distributions . . . . . . . 7.3

multivariate Gaussian distribution . . . . . 6.4.8, 9.4.3.1.3assignment to a vector quantity . . . 6.4.8.1, 9.4.1.4expectation and covariance matrix . . . . . . . . 6.4.8.3probability density function . . . . . . . . . . . . . . . 6.4.8.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . 6.4.8.4, C.5

mutual uncertainty. . . . . . . . . . . . . . . . . . . . . . . . .3.11, 5.6.3

Nnormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4numerical tolerance3.20, 5.1.3, 7.2.1, 7.2.3, 7.9.1, 7.9.2,

7.9.4, 8.1.3, 8.2, 9.1.2, 9.2.2.3, 9.2.2.4, 9.2.2.7,9.2.4.2, 9.2.4.5, 9.3.2.2, 9.3.2.6, 9.5.3.2

Ooutput quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1, 5.1.1

Ppartial derivative . . . 5.6.3, 5.8.1, 5.11.4, 5.11.6, 9.3.1.3,

9.3.2.5, 9.4.2.2.1, 9.4.2.2.4, B.1, F.3.1.2PDF . . . . . . . . . . see probability density function (PDF)Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.11.1principle of maximum entropy . . . . . . 6.1.1, 6.3, 6.4.2.1,

6.4.3.1, 6.4.6.1, 6.4.7.1, 6.4.10.1probability . . . . . . . . 3.1, 3.2, 5.1.2, 5.11.2, 7.5.1, 9.2.2.8,

9.4.2.2.9, 9.4.2.2.11, 9.5.2.9.1, 9.5.2.10.1, D.2probability density function (PDF) . . 3.3, 4.2, 4.3, 4.15

asymmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4, 5.3.6for input quantities5.1.1, 5.1.2, 5.4.4, 5.6.2, 5.11.2,

5.11.4, 6, 6.1.3, 6.1.5, 6.2–6.4, 7.3, 7.4.1, 7.6,7.8.1

for output quantity . . . . . D, 5.1.1, 5.2, 5.4.4, 5.6.2,5.7.2, 5.11.2, 5.11.4, 7.2.1, 7.5.2, B.1

joint . . . 4.4, 4.5, 5.1.1, 6.1.2, 6.1.4, 6.4.8.2, 6.4.8.4,7.3, 7.8.1, 9.4.1.8

sampling from . . . . . . . . . . . . . . . . . . . . . C, 7.3, 9.2.2.4symmetric . . . . . . . . . . . . . . . . . . . . 5.3.3, 5.3.5, 9.2.2.8

probability distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1from previous uncertainty calculations . . . . . . . . 6.5joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1multivariate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1sampling from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C, 7.3univariate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1

probability element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3propagation . . . . . . see stages of uncertainty evaluationpropagation of distributions . . . . . . . 3.17, 5.2, 5.4, 5.9.6

analytical . . . . . . . . . . . . . . . .5.4.1, 9.4.2.1.1, 9.4.2.2.2implementations of . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4

propagation of uncertainty . . . . . . . . . . . . 3.18, 4.9, 5.4.1,5.6.2, 5.7.1, 5.8.1, 5.11.2, 5.11.6, 7.4.2, 8.1.2,9.2.2.3, 9.3.1.3, 9.4.2.2.1, 9.4.2.2.8

Rrandom variable . 3.1, 4.1, 4.3, 5.6.3, 5.9.6, 6.2.1, 6.2.2,

7.9.4rectangular distribution. . . . . .6.4.2, 9.2.3, 9.2.4, C.2, E

assignment to a quantity.6.4.2.1, 9.3.1.4, 9.5.2.6.2expectation and variance . . . . . . . . . . . . . . . . . . 6.4.2.3probability density function . . . . . . . . . . . . . . . 6.4.2.2


JCGM 101:2008

sampling from. . . . . . . . . . . . . . . . . .6.4.2.4, 9.1.4, C.3rectangular distribution with inexactly prescribed lim-

its. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.3assignment to a quantity . . . . . . . . 6.4.3.1, 9.5.2.9.2,

9.5.2.10.2expectation and variance . . . . . . . . . . . . . . . . . . 6.4.3.3probability density function . . . . . . . . . . . . . . . 6.4.3.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.3.4

rectangular number generatorquality of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C.3.1.1recommended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3.3

reliability of uncertainty . . . . . see standard uncertaintyreporting the results . . . . . . . . . . . . . . . . . . . . . . 1, 5.5, 9.1.3

Ssensitivity coefficients . . . . . . 5.4.3, 5.6.3, 5.11.4, 5.11.6,

9.3.2.5, 9.5.2.6.2, Bsignificant decimal digits . . .1, 3.20, 4.13, 5.5.2, 6.4.3.4,

7.2.1, 7.6, 7.9.2, 8.1.3, 8.2, 9.1.3skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1, 9.4.2.4.1sorting algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1, 7.8.2stages of uncertainty evaluation . . . . . . . . . . . 5.1.1, 5.6.1

formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1, 5.1.3propagation5.1.1, 5.1.3, 5.4, 5.4.3, 5.6.1, 5.6.3, 5.9summarizing . . . . . 5.1.1, 5.1.3, 5.3, 5.6.1, 5.6.3, 5.9

standard deviation . . . . . . . . 3.8, 5.1.1, 5.3.1, 5.6.2, 5.6.3of a set of indications . . . . . . . . . . . . . . . . . . . . . . 5.11.2of sampled model values . . . . . . . . . . . . . . . . . . . . . . 7.6pooled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.9.6

standard uncertainty . . . . 4.10, 5.1.1, 5.1.2, 5.3.1, 5.5.1,5.11.2, 5.11.5, 5.11.6, 8.1.3

from GUM uncertainty framework . . . . 5.6.2, 5.6.3from Monte Carlo method . . . . . . . . . . . . . . 5.9.6, 7.6reliability of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.9.4

summarizing . . . . . . see stages of uncertainty evaluation

Tt-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5, 6.4.9

assignment to a quantity 6.4.9.2, 6.4.9.7, 9.5.2.2.2,9.5.2.3.2, 9.5.2.4.2, 9.5.2.5.2

expectation and variance . . . . . . . . . . . . . . . . . . 6.4.9.4probability density function . . . . . . . . . . . 3.5, 6.4.9.3sampling from. . . . . . . . . . . . . . . . . . . . . . . . 6.4.9.5, C.6

Taylor series approximation . . . 4.9, 5.4.1, 5.6.2, 5.11.4,8.1.2, 9.1.1, 9.1.6, B.1, F.3.1.3

tests of randomness . . . . . . . . . . . . . . . . . . . . . . . 7.3, C.3.2.2trapezoidal distribution. . . . . . . . . . . . . . . . . . . . . . . . . .6.4.4

assignment to a quantity . . . . . . . . . . . . . . . . . . 6.4.4.1expectation and variance . . . . . . . . . . . . . . . . . . 6.4.4.3probability density function . . . . . . . . . . . . . . . 6.4.4.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.4.4

triangular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5assignment to a quantity . . . . . . . . . . . . . . . . . . 6.4.5.1expectation and variance . . . . . . . . . . . . . . . . . . 6.4.5.3probability density function . . . . . . . . . . . . . . . 6.4.5.2sampling from. . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.5.4

Type A and Type B evaluations of uncertainty . . 5.1.2,5.11.2, 5.11.4, 6.1.3, 6.4.9.4

UU-shaped distribution . . . . . . . . see arc sine distributionuncertainty evaluation problems. . . . . . . . . . . . . . . . . . . . .1uncertainty matrix . . . . . . . . . . . . . . . 3.11, 6.4.8.1, F.3.2.1

Vvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7, 4.3, 5.3.1

of a set of indications . . . . . . . . . . . . . . . . . . . . . 6.4.9.2variance-covariance matrix . . . . . . . . . . . . . . . . . . . . . . . 3.11VIM see International vocabulary of basic and general

terms in metrology (VIM)

WWelch-Satterthwaite formula . . . . . . . 5.6.3, 5.7.2, 5.11.6,

6.4.9.4, 9.5.3.1


evaluation of measurement data

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